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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| /■ary Sytttm ity of Wisconsin-Maditoil otate Street dison, Wl 53706-1484 S.A. A MANUAL OF RULES, TABLES, AND DAT4 FOR MECHANICA^ lENGINEERS BASED ON THE MOST RECENT INVESTIGATIONS: OF CONSTANT USE IN CALCULATIONS AND ESTIMATES RELATING TO Stkskcth of Matbkials AMD OP Elementary Constructions; Labour; Heat and its Applications, Steam and its Properties, Combustion and Fuels, Steam Boilers, Steam Engines, Hot-Air Engines, Gas-Engines ; Flow op Air and op Water; Air Machines; Hydraulic Machines; Mill-gearing; Friction and the Resistance of m achinbry, &c ; weights, measures, and monies, british and forbign, with the reciprocal Equivalents for the Conversion of British and French Compound Units op Weight, Pressure, Time, Space, and Moi^BV : Specific Gravity and ^ THE Weight op Bodies; Weight op Metals, &c. with Tables of Logarithms, Circles, Squares, Cubes, Square-Roots, and Cube-Roots; AND MANY OTHER USEFUL MATHEMATICAL TaBLBS. BY DANIEL KINNEAR CLARK, % I AUTHOR OF MEMBER OF THE INSTITUTION OP CIVIL ENGINEERS; RAILWAY MACHINERY," "EXHIBITED MACHINERY OF 1862," BTC. SECOND EDITION. * LONDOVsT: BLACKIE & SON: PATERNOSTER BUILDINGS, GLASGOW AND EDINBURGH. 1878. A/i Rights Reset ved. QhABQOWl W. O. BLAOKIS AJSD CO., PBINTEBS, VILLAFIXLD. PREFACE. This Work is designed as a book of general reference for Engineers : — ^to give within a moderate compass the leading rules and data, with numerous tables, of constant use in calculations and estimates relating to Practical Mechanics. The Author has endeavoured to concentrate the results of the latest investigations of others as well as his own, and to present the best information, with perspicuity, conciseness, and scientific accuracy. Amongst the new and original features of this Work, the follow- ing may be named : — In the section on Weights and Measures, the weight, volume, and relations of water and air as standards of measure, are concisely set forth. The various English measures, abstract and technical, are given in full detail, with tables of various wire-gauges in use: and equivalent values of compound units of weight, power, and measure — as, for example, miles per hour and feet per second. The French Metric Standards are defined, according to the latest determinations, with tables of metric weights and measures, equi- valents of British and* French weights and measures, and a number of convenient approximate equivalents. There is, in addi- tion, a full table of equivalents of French and English compound units of weight, pressure, time, space, and money — as, for example, pounds per yard and kilogrammes per metre; which will be found of great utility for the reciprocal conversion of English and French units. The tables of the Weight of bars, tubes, pipes, cylinders, plates, sheets, wires, &c., of iron and other metals, have been calculated expressly for this Work, and they contain several new features designed to add to their usefulness. They are accompanied by a summary of the various units of weight of wrought iron, cast iron, and steel, with plain rules for the weight. In the section on Heat and its Applications, the received mechan- ical theory is defined and illustrated by examples. The relations of the pressure, volume, and temperature of air and other gases. vi PRfeFACE. with their specific heat, are investigated in detail. The transmission of heat through plates and pipes, between water and water, steam and air, &c., for purposes of heating or cooling, is verified by many experimental data, which are reduced to units of performance. The physical properties of steam are deduced from the results of Regnault's experiments, with the aid of the mechanical theory of heat A very full table of Ihe Properties of Saturated Steam is given. The table is, for the most part, reproduced from the article ** Steam," contributed by the Author to the Encyclopedia Britannicay 8th edition, and it was the first published table of the same extent, in the English language, based on Regnault's data. An original table of the properties of saturated mixtures of air and aqueous vapour is added. In the section on Combustion, new and simple formulas and data are given for the quantity of air consumed in combustion, and of the gaseous products of combustion, the heat evolved by combus- tion, the heating power of combustibles, and the temperature of combustion ; with several tables. On Coal as a Fuel, both English and Foreign, its composition, with the results of many series of experiments on its combustion, are collected and arranged. Thequantityof air consumed in its com- bustion, and of the gaseous products, with the total heat generated, are calculated in detail. Coke, lignite, asphalte, wood, charcoal, peat, and peat-charcoal, are similarly treated ; whilst the combus- tible properties of tan, straw, liquid-fuels, and coal-gas, are shortly treated. The section on Strength of Materials is wholly new. The great accumulation of experimental data has been explored, and the most important results have been abstracted and tabulated. The results of the experiments of Mr. David Kirkaldy occupy the greater por- tion of the space, since he has contributed more, probably, than any other experimentalist to our knowledge of the Strength of Materials. The Author has investigated afresh the theory of the transverse strength and deflection of solid beams, and has deduced a new and simple series of formulas from these investigations, the truth of which has been* established with remarkable force by the evidence of experi- ment These investigations, based on the action of diagonal stress, throw light upon the element called by Mr. W. H. Barlow, "the resistance of flexure:" revealing, in a simple manner, the nature of that hitherto occult entity; and showing that flexure is not the cause, but the effect of the resistance. In addition to formulas PREFACE. vil for beams of the ordinary form, special formulas have been deduced for the transverse strength and deflection of railway rails, double-headed or flanged, of iron or steel; in the estab- lishment of which he has availed himself of the important experimental data published by Mr. R. Price Williams, and by Mr. B. Baker. To our knowledge of the strength of timber, Mr. Thomas Laslett has recently made important additions, and the results of his experiments have been somewhat fully abstracted and analyzed. But woods, by their extremely variable nature, are not amenable, like wrought-iron and steel, to the unconditional applica- tion of formulas for transverse strength. The Author has, never- theless, deduced from the evidence, certain formulas for the trans- verse strength and deflection of woods, with tables of constants, which, if applied with intelligence and a knowledge of the uncer- tainties, cannot fail to prove of utihty. The Torsional Strength of Solid Bodies has also been investigated afresh, and reduced to new formulas. In dealing with the Strength of Elementary Constructions, the Author has brought together many important experimental results. In treating of rivet-joints and their employment in steam-boilers, he has, he believes, clearly developed the elements of their strength and their weakness. By a close comparison of the results of tests of cast-iron flanged beams, it is plainly shown that the ultimate strength of a cast-iron beam is scarcely affected by the proportionate size of the upper flange, and that the lower flange' and the web are, practically, the only elements which regulate the strength. The tests of solid-rolled and rivetted wrought-iron joists are also ana- lyzed ; and for the strength and deflection of these, as for those of cast-iron flanged beams, new and simple rules and formulas are given, A new investigation, with appropriate formulas, is given for the bursting strength of hollow cylinders, of whatever thickness. It is shown that the variation of stress throughout the thickness, follows a diminishing hyperbolic ratio from the inner surface to- wards the outer surface. The resistance of tubes and cylindrical flues to collapsing pressure is also investigated, and formulas based on the results of experience are given. On the subject of Mill-gearing, a new and compact table of the pitch, number of teeth, and diameter of toothed wheels is given, with new formulas and tables for the strength and horse-power of the teeth of wheels, and for the weight of toothed wheels. New formulas and tables are given for the driving power of leather viii • PREFACE. belts, and the weight of cast-iron pulleys. For the strength of Shafting, — cast-iron, wrought-iron, and steel, — a new and complete series of formulas has been constructed, comprising its resistance to transverse deflection and to torsion, with very full tables of the weight, strength, power, and span of shafting. The Evaporative Performance of Steam-boilers is exhaustively investigated with respect to the proportions of fuel, water, grate- area, artd heating-surface, and the relations of these quantities are . reduced to simple formulas for different types of boilers. The actual evaporative performances of boilers are abstracted in tabular form, comprising those of the Lancashire and the Galloway boilers at Wigan, tested by Mr. Lavington E. Fletcher. The Performance of Steam worked expansively, in single and in compound cylinders, is exhaustively analyzed by the aid of diagrams ; the similarity and the dissimilarity of its action in the Woolf-engine and the Receiver-engine, are investigated; and the principles of calculation to be applied respectively to these, the leading classes of compound engines, are explained. The best working ratios of expansion are deduced from the results of numerous experiments and observations on the performance of steam-engines. The principles of Air-compressing Machines, and Compressed-air Engines, — a branch of mechanical practice of comparatively recent origin, — are investigated, and convenient formulas and tables for use are deduced. Such are some of the new features of this volume. It may be added, that the other portions of the Work, likewise, have been carefully elaborated. The whole of the materials for its preparation have been drawn from the best available sources, foreign as well as English. Vast stores of the results of experience are accumulated in the Proceedings of t/te Institution of Civil Engifieers^ the Proceedings of tfie Institution of Mechanical Engineers, as well as in various periodicals and journals English and foreign. From these and other acknowledged sources, the Author has drawn much of his material. D. K. CLARK. 8 Buckingham Street, Addphi, London, 20th March, 1877. CONTENTS. GEOMETRICAL PROBLEMS. . PACB Straight Lines — Straight Lines and Circles— Circles and Rectilineal Figures — The Ellipse — The rarabola — The Hyperbola — The Cycloid and Epicycloid — The Catenary — Circles — Plane Trigonometry — Mensuration of Surfaces — Solids — Heights and Distances, I MATHEMATICAL TABLES. Explanation of the following Tables : — 32 Logarithms of Numbers from i to 10,000, 38 Hyperbolic Logarithms of Numbers from 1. 01 to 30, 60 Numbers or Diameters of Circles, Circumferences, Areas, Squares, Cubes, Square Roots, and Cube Roots, .66 Circles : — Diameter, Circumference, Area, and Side of Equal Square, . 87 Lengths of Circular Arcs from 1° to iSo**, 95» 97 Areas of Circular Segments, 100 Sines, Cosines, Tangents, Cotangents, Secants, and Cosecants of Angles, . . 103 Logarithmic Sines, Cosines, Tangents, and Cotangents of Angles, . . 1 10 Rhumbs, or Points of the Compass, 117 Reciprocals of Numbers from I to 1000, - 118 WEIGHTS AND MEASURES. Water as a Standard — Weight and Volume of pure Water — The Gallon and other Measures of Water — Pressure of Water — Sea-water — Ice and Snow — French and English Measures of Water, 1 24 Air as a Standard — Pressure of the Atmosphere — Measures of Atmospheric Pres- sure — ^Weight of Air — Volume — Specifie Heat, 127 Great Britain and Ireland — Imperial Weights and Measures, .128 Measures of Length : — Lineal — Land — Nautical— Cloth, 129 Wire-gauges, 130 Inches and their Equivalent Decimal Values in part^of a Foot — Fractional Parts of an Inch, and their Decimal Equivalents, 135 Measures of Surface : — Superficial — Builders' Measurement— Land, . .136 Measures of Volume : — Solid or Cubic — Builders* Measurement, . • *37 Table of Decimal Parts of a Square Foot in Square Inches, . . .138 Measures of Capacity : — Li<|uid— Dry — Definition of the Standard Bushel — Coal — Old Wme and Spirit — Old Ale and Beer — Apothecaries* Fluid, . 138 Measures of Weight: — Avoirdupois — Troy — Diamond — Apothecaries* — Old Apothecaries' — Weights of Current Coins — Coal— Wool — Hay and Straw — Com and Flour, 140 Miscellaneous Tables : — Drawing Papers — Commercial Numbers — Stationery — Measures relating to Building — Commercial Measures — Measures for Ships, 143 Comparison of English Compound Units : — ^Measures of Velocity — Of Volume and Time — Of Pressure and Weight — Of Weight and Volume — Of Power, 144 X CONTENTS. PAGE France — The Metric Standards of Weights and Measures — Metre— Kilogramme, . 146 Countries where the system is legalized, . . . . . • « • .146 Measures of Length, 147 Wire-gauges, 148 Measures of Surface, 149 Measures of Volume : — -Cubic — Wood, 149 Measures of Capacity : — Liquid — Dry, 149 Measures of Weight, 150 Equivalents of British Imperial and French Metric Weights and Measures, . 150 Measures of Length — Tables of Equivalent Values of Millimetres and Inches — Square Measures or Measures of Surface — Cubic Measures — Wood Mea- sure — Measures of Capacity — Measures of Weight, 150 Approximate Equivalents of English and French Measures, 156 Equivalents of French and English Compound Units of Measurement : — Weight, Pressure, and Measure — Volume, Area, and Length — Work — Heat — Speed — Money, 157 German Empire : Weights and Measures : — Length — Surface — Capacity — Weight, 160 Values of the German Fuss or Foot in the various States, 161 Old Weights and Measures in Prussia (Kingdom of) — Bavaria (Kingdom of) — Wiirtembeig (Kingdom of) — Saxony (Kingdom of) — Baden (Grand-duchy of) — Hanse ']u)wns: — Hamburg — Bremen — Lubec — Old German Customs Union — Oldenburg — Hanover, &c., 162 Austrian Empire, 170 Russia, 171 Holland — Belgium — Norway and Denmark — Sweden 173 Switzerland — Spain — Portugal — Italy 175 Turkey — Greece and Ionian Islands — Malta, 178 Egypt — Morocco — Tunis — Arabia — Cape of Good Hope, 179 Indian Empire — Bengal — Madras — Bombay — Ceylon, 180 Burmah — China— Cochin-China — Persia — ^Japan — ^Java, 183 United States of North America, 186 British North America, 187 Mexico — Central America and West Indies — West Indies (British) — Cuba — Guate- mala and Honduras — British Honduras — Costa Rica — St. Domingo, . . 187 South America — Colombia — Venezuela — Ecuador — Guiana — Brazil — Peru — Chili — Bolivia — Argentine Confederation — Uruguay — Paraguay, .... 188 Australasia : — New South Wales — Queensland — Victoria — New Zealand, &c., . 189 MONEY— BRITISH AND FOREIGN. Great Britain and Ireland : — Value, Material, and Weight of Coins — Mint Price of Standard Gold, &c., 190 France: — Material and Weight of French Coins, and Value in English Money, . 190 German Empire : — Names and Equivalent Values of Coins, 191 North and South Germany (Old Currency of), 191 Hanse Towns (Old Monetary System of): — Hamburg, Bremen, Lubec, . . . 191 Austria — Russia — Holland — Belgium — Denmark — Sweden — Norway, . . . 192 Switzerland — Spain — Portugal — Italy — Turkey — Greece and Ionian Islands — Malta, 1 93 Egypt — Morocco — Tunis — Arabia — Cape of Good Hope, 194 Indian Empire — China — Cochin-China — Persia — ^Japan — Java, .... 195 United States of North America, 195 Canada — British North America, 196 Mexico— West Indies (British) — Cuba — Guatemala — Honduras — Costa Rica — St. Domingo, 196 CONTENTS. XI PACK Sooth America — C olombia — Venezuela — Ecuador — Guiana — BrazU — Peru — Ch ili — Bolivia — ATgentine Confederation — Uniguay — Paraguay, . . . igt Austialasia, . . - 197 WEIGHT AND SPECIFIC GRAVITY. Standard Bodies and Temperatures for Comparative Weight — Rules for Specific Gravity, u 198 General Comparison of the Weights of Bodies, •199 Tables of the Volume, Weight, and Specific Gravity of Metallic Alloys — Metals — Stones, 200 Coal — Peat — Woods — Wood-Charcoal,* 206 Animal Substances — Vegetable Substances, • . 212 Weight and Volume of various Substances, by Tredgold, 213 Weight and Volume of Goods carried over the Bombay, Baroda, and Central Indian Railway, 213 Weight and Specific Gravity of Liquids, . 215 Weight and Specific Gravity of Gases and Vapours, 216 WEIGHT OF IRON AND OTHER METALS. Data for Wrought Iron — for Steel— for Cast Iron, . .' . . . .217 Tables of Weights: — Weights of given Volumes of Metals — Volumes of given Weights of Metals — Weight of One Square Foot of Metals — Weight of Metals of a given Sectional Area, 218 Special Tables for the Weight of Wrought Iron: — Rales for the Weight of Wrought Iron — Cast Iron — and Steel, , . . 223 Rule for the Length of I cwt. of Wire of different Metals, of a given thickness, 224 Weight of French Galvanized Iron Wire, 225 Special Tables of the Weight of Wrought-Iron Bars, Plates, &c. ; Multipliers for other Metals : — Flat Bar Iron — Square Iron — Round Iron — Angle Iron and Tee Iron — Wrought-Iron Plates — Sheet Iron — Black and Galvanized- Iron Sheets — Hoop Iron — Warrington Iron Wire — Wrought-Iron Tubes, by Internal Diameter — Wrought-Iron Tubes, by External Diameter, . 226 Weight of Cast Iron, Steel, Copper, Brass, Tin, Lead, and Zinc — Special Tables : — Cast-Iron Cylinders, by Internal Diameter — Cast- Iron Cylinders, by External Diameter — Volumes and Weight of Cast-Iron Balls, for given Diameters; Multipliers for other Metals — Diameter of Cast- Iron Balls for given Weights, 253 Weight of Flat-Bar Steel — Square and Round Steel — Chisel Steel, . . . 259 Weight of One Square Foot of .Sheet Copper — Copper Pipes and Cylinders, by Internal Diameter — Brass Tubes, by External Diameter — One Square Foot of Sheet Brass, 261 Siie and Weight of Tin Plates — Weight of Tin Pipes and Lead Pipes — Dimen- sions and Weight of Sheet Zinc, 268 FUNDAMENTAL MECHANICAL PRINCIPLES. Forces in Equiubrium:— Solid Bodies — Fluid Bodies, 271 Motion : — Uniform Motion — Velocity — Accelerated and Retarded Motion, . . 277 Gravity; — Relations of Height, Velocity, and Time of Fall — Rules and Tables, . 277 Accelerated and Retarded Motion in General: — General Rules— Descent on Inclined Planes, 282 Mass, 287 Mechanical Centres :— Centre of Gravity — Centre of Gyration— Radius of Gyration — Moment of Inertia — Centre of Oscillation — The Pendulum — Length of Seconds Pendulum — Centre of Percussion 287 Central Forces : — Centripetal Force — Centrifugal Force, 294 Xll CONTENTS. PAGB Mechanical Elements r—The Lever— The Pulley— The Wheel and Axle— The Inclined Plane — Identity of the Inclined Plane and the Lever — The Wedge — The Screw, 296 Work:— English and French Units of Work — Work done by the Mechanical Ele- ments — By Gravity — Work accuxmilated in Moving Bodies — Work done by Percussive Force, 312 HEAT. Thermometers: — Table of Equivalent Degrees by Centigrade and Fahrenheit — Pyrometers, 317, 967 Movements of Heat: — Radiat;ion — Conduction — Convection, .... 329 The Mechanical Theory of Heat: — Mechanical Equivalent of Heat— Joule's Equivalent in English and French Units — Illustrations, .... 332 Expansion by Heat: — Linear and Cubical Expansion, 335 • Table of Linear Expansion of Solids, 336 Expansion of Liquids,^ . 338 Expansion of Gases — The Absolute Zero-point — Table of the Compression of Gases by Pressure under a Constant Temperature, 342 Relations of the Pressure, Volume, and Temperature of Air and other Gases — General Rules — Special Rules for One round weight of a Gas, with Table of Coefficients — Table of the Volume, Density, and Pressure of Air at various Temperatures, . * . . . 346 Specific Heat: — Specific Heat of Water, with Table— Specific Heat of Air- Specific Heat of Solids — Specific Heat of Liquids — Specific Heat of Gases, . 352 Fusibility or Melting Points of Solids: — Table, 363 Latent Heat of Fusion of Solid Bodies, with Rule and Table, .... 367 Boiling Points of Liquids, 368 Latent Heat and Total Heat of Evaporation of Liquids, 370 Boiling Points of Saturated Vapours under various Pressures, . • 37^ Latent Heat and Total Heat of Evaporation of Liquids under One Atmosphere, 372 Liquefaction and Solidification of Gases, 372 Sources of Cold: — Siebe's Ice- making Machine — Carre's Cooling Apparatus — Frigorific Mixtures, 373 STEAM. Physical Properties of Steam, 378 Gaseous Steam — Its Expansion — Its Total Heat, 383 Specific Heat of Steam — Specific Density of Steam — Density of Gaseous Steam, 384 Properties of Saturated Steam from 32' to 212* F., 386 Properties of Saturated Steam for Pressures of from I lb. to 400 lbs., . 387 Comparative Density and Volume of Air and Saturated Steam, . • 39^^ MIXTURE OF GASES AND VAPOURS. Respective Pressures of Gas and Vapours in Mixture, 392 Hygrometry, 392 Properties of Saturated Mixtures of Air and Aqueous Vapour, with Table, . 394 COMBUSTION. Combustible Elements of Fuel — Process of Combustion, 398 Air Consumed in the Combustion of Fuels :— Quantity of the Gaseous Pro- ducts of the Complete Combustion of One Pound of Fuel — Surplus Air, . 400 Heat Evolved by the Combustion of Fuel : — Heat of Combustion of Simple and Compound Bodies — Heating Powers of Combustibles, .... 402 Temperature of Combustion, 407 CONTENTS. xm FUELS. PAGE Pods or Combustibles generally used, . « 409 Coal: — Its Varieties — Small Coal: — Its Utilization — Washing Small Coal — Deterioration of Coal by Exposure to Atmosphere, 409 British Coals — Composition of Bituminous Coals — Dr. Richardson's Analysis, . 412 Weight and Composition of British and Foreign Coals, by Delab^che and Playfair, 413 Weight and Bulk of British Coals, 416 Hygroscopic Water in British Coals, 416 Torbanehill or Boghead Coal, with Table of its Composition, . . . > 417 American and Foreign Coals : — Composition, Weight and Bulk, . . .418 French Coals: — Utilization of the Small Coal — Composition of French Coals — Mean Density, Composition, and Heating Power, 420 Indian Coals : — Australian and Indian Coals — Composition, ..... 423 Combustion of Coal : — Process of Combustion — Gaseous Products of the Com- bustion of Coal — Surplus Air — Total Heat of Combustion of British Coals, . 426 CoK£ : — Proportion of Coke from Coals — Anthracitic Coke — Weight and Bulk of Coke — Composition of Coke — Moisture in Coke — Heating Power of Coke, . 430 LiCNiT£ AND Asphalts : — Density, Composition, and Heating Power of Lignites and Asphaltes, 436 Wood: — Moisture in Wood — Composition — Weight and Bulk of Wood, with Table — Firewood— Quantity of Air Chemically Consumed in the Complete Combustion of Wood — Gaseous Products — Total Heat of Combustion — Temperature of Combustion, . . 439 Wood-Charcoal : — ^Yield of Charcoal — Composition, with Table of Composition at various Temperatures — Carbonization of Wood in Stacks, and Yield of Charcoal — Manufacture of Brown Charcoal — Distillation of Wood — Oiiarbon de Paris (artificial fuel) — Weight and Bulk of Wood-Charcoal — Absolute Density of Charcoal — Moisture in Charcoal — Air Consumed in the Combus- tion of Charcoal — Gaseous Products — Heat of Combustion, . . . 444 Peat: — Nature and Composition — Condensed Peat — Average Composition — Pro- ducts of Distillation — Heating Power of Irish Peat, 452 Peat-Charcoal: — Composition and Heating Power, 455 Tan : — Composition and Heating Power, , 455 Straw: — Composition, 456 Liquid Fuels : — Petroleum, Petroleum-Oils, Schist Oil, and Pine-wood Oil ; their Composition and Heating Power, 456 Coal-Gas : — Composition and Heating Power, . . . ... • • 457 APPLICATIONS OF HEAT. Transmission of Heat through Solid Bodies — from Water to Water THROUGH Solid Plates and Beds: — M. Peclet's Experiments— Mr. James R. Napier's Experiments — Circumstances which affect the Ratio of Trans- misaon — Mr. Craddock's Experiments, 459 Heating and Evaporation of Liquids by Steam through Metallic Surfaces: — Experiments by Mr. John Graham, by M. Clement, by M. Pcclet, by MM. Laurens and Thomas, by M. Havrez, by Mr. William Anderson, by Mr. F. J. Bramwell — Table of Performance of Coiled Pipes and Boilers in Heating and Evaporating Water by Steam, with Deductions, 461 CoouNG of Hot Water in Pipes: — Observations of M. Darcy — Experiments by Tredgold — Deductions, 469 Cooling of Hot Wort on Metal Plates in Air: — Results of Experiments at Trueman*s Brewery, 470 CoouNG OF Hot Wort by Cold Water in Metallic Refrigerators: — Table of Results of Performance, and Deductions, 471 XIV coNTE^r^s. PAGE Condensation of. Steam in Pipes Exposed to AiR:—Expcriments by Tred- gold, and by M. Burnat, on Pipes with various Coverings, with Table — Experiments by Mr. B. G. Nichol, by M. Clement, by M. Grouvelle — Condensation of Steam in a Boiler Exposed in Open Air, .... 472 Condensation of Vapours in Pipes or Tubes by Water:— M. Audenet's Experiments on Steam — Mr. B. G. Nichol's Experiments — Condensation of other Vapours, 475 Warming AND Ventilation: — Allowance of Air for Ventilation, . . . 477 Ventilation of Mines by Heated Columns of Air.— Furnace- Ventilation — Mr. Mackworth's Data 479 Cooling Action of Window-Glass:— Mr. Hood's Data, .... 480 Heating Rooms by Hot Water: — Mr. Hood's Estimates — Total Quantity of Air to be Warmed per Minute — Table of the Length of 4-inch Pipe required to Warm any Building — Boiler-power — French Practice — Perkins' System, . 481 Heating Rooms by Steam: — Length of 4-inch Pipe required — French Practice, 486 Heating by Ordinary Open Fires and Chimneys:— M. Claudel's Data, . 488 Heating by Hot Air and Stoves: — Sylvester's Cockle-Stove — French Prac- tice — House- Stoves placed in the Rooms to be Warmed — House- Stoves placed outside the Rooms to be W^armed, 488 Heating of Water by Steam in Direct Contact: — Mr. D. K. Clark's Experiments, 490 Evaporation (Spontaneous) in Open Air: — Dalton's Experiments, and Detluc- tions — Rule for Spontaneous Evaporation — Dr. Pole's Formula, . . '491 Desiccation by Dry Warm Air: — Design of a Diying Chamber— Results of Experiments — Drying-house for Calico — Drying Linen and Various Stuffs — Drying Stuffs by Contact with Heated Metallic Surfaces — Drying Grain — Drying Wood, 493 Heating of Solids: — Cupola Furnace — Plaster Ovens — Metallurgical Furnaces — IMast Furnaces, 497 STRENGTH OF MATERIALS. Definitions 500 Work of Resistance of Material, 501 Coefficient of Elasticity, 503 Transverse Strength of Homogeneous Beams, 503 Symmetrical Solid Beams: — Investigation and Generalized Formula, . . 503 Formula for the Transverse Strength of Solid Beams of Symmetrical Section, without Overhang, and Flanged or Hollow — For Unsymmetrical Flanged Beams — Neutral Axis — Elastic Strength, 509 Forms of Beams of Uniform Strength:— Semi-Beams Loaded at One End I — Uniformly Loaded, 517 Forms of Beams of Uniform Strength, Supported at Both Ends — Under a Con- centrated Rolling Load, 521 Shearing Stress in Beams and Plate-Girders 525 Deflection of Beams and Girders :— Investigation — Rectangular Beams— Double-flanged — Uniform Beams Supported at Three or more Points, . . 527 Torsional Strength of Shafts:— Round— Hollow— Square— Deflection, . 534 Strength of Timber:— Results of Experiments, 537 Transverse Strength of Timber of Large Scantling, 542 Elastic Strength and Deflection of Timber: — Experiments by MM. Chevandier and Wertheim, by Mr. Laslett, by Mr. Kirkaldy, by Mr. Barlow, . . 545 Rules for the Strength and Deflection of Timber, 548 Strength of Cast Iron: — Tensile Strength and Compressive Strength- Results of Experiments, 553 Shearing Strength, 561 CONTENTS. XV PAGB Tmnsvcrse Strength: — Results of Experiments — Test Bars — Transverse Deflection and Elastic Strength, . . / 561 Torsional Strength, 565 Strekgth of Wrought Iron :~Tensile Strength, &c.— Mr. Kirkaldy's Experi- ments, 567 Experiments of the Steel Committee of Civil Engineers, 579 Hammered Iron Bars (Swedish) — Knipp and Yorkshire Plates— Pi-ussian Plates, 581 Iron Wire, 586 Shearing and Punching Strength, 5S7 Transverse Strength — Deflection and Elastic Strength, 588 Torsional Strength, 590 Strength of Steel:— Mr. Kirkaldy*s Early Experiments— Hematite Steel— Krupp Steel, 593 Experiments of the Steel Committee, 596 Experiments at H.M. Gun Factory, Woolwich — Fagersta Steel, Mr. Kirkaldy's Experiments, in seven series, 604 Siemens- Steel Plated and Tyres — Mr. Kirkaldy's Experiments, . . .612 Whitworth's Fluid- compressed Steel, 614 Sir Joseph Whitworth's Mode of Expressing the Value of Steel, . .615 ChemofTs Experiments on Steel, 616 Steel Wire, 617 Shearing Strength of Steel, 617 Transverse Strength and Deflection, 617 Torsional Strength, 619 Strength Relatively to the Proportion of Constituent Carbon, . . . .621 Resistance of Steel and Iron to Explosive Force > . 622 Recapitulation of Data on the Direct Strength of Iron and Steel: — Tensile and Compressive Strength of Cast Iron, Wrought Iron, and Steel — Diagram of the Relative Elongation of Bars of Cast Iron, Wrought Iron, and Steel, 623 Working Strength of Materials— Factors of Safety: — Factors of Safety for Cast Iron, Wrought Iron, Steel, and Timber — Load on Foundations, Mason-work — Ropes — Dead Load — Live Load, 625 Tensile Strength of Copper and other Metals:— Tables of the Strength of Copper and its Alloys: Tin, Lead, Zinc, Solder, 626 Tensile Strength of Wire of Various Metals:— Tenacity of Metallic Wires at Various Temperatures — Wires of Various Metals, . . 628 Strength op Stone, Bricks, &c. : — ^Table of the Tensile Strength of Sandstones and Grits, Marbles, Glass, Mortar, Plaster of Paris, Portland Cement, Roman Cement, Granites, Whinstone, Limestone, Slates, Bricks, Brickwork in Cement — Adhesion of Bricks, 629 STRENGTH OF ELEMENTARY CONSTRUCTIONS. Rivet-Joints: — In Iron Plates, 633 In Steel Plates, 642 PiLUiRS or Columns : — Compressive Strength 643 Cast-Iron Flanged Beams: — Transverse Strength, 647 Deflection and Elastic Strength, 652 Wrought-Iron Flanged Beams or Joists: — Solid Wrought-iron Joists — Transverse Strength and Deflection, 653 Riretted Wrought-iron Joists, 657 Bt'CKLED Iron Plates, 660 I XVI CONTENTS. PACB Railway Rails: — Transverse Strength of Rails of SynMnetrical Section, . .661 Rails of Unsymmetrical Section, . 665 Deflection of Rails, 668 Steel Springs: — ^Laminated and Helical, 671 Ropes: — Hemp and Wire, 673 Chains, 677 Leather Belting, 679 Bolts and Nuts 680 Screwed Stay-Bolts and Flat Surfaces 685 Hollow Cylinders — Tubes, Pipes, Boilers, &c.: — Resistance to Internal or Bursting Pressure — Transverse Resistance, 687 Longitudinal Resistance to Bursting Pressure, 692 Wrought-iron Tubes, 693 Cast-iron Pipe 693 Resistance to External or Collapsing Pressure — Solid- drawn Tubes — Large Flue Tubes — Lead Pipes, 694 Framed Work — Cranes, Girders, Roofs, &c.: — The Triangle the Funda- mental Feature, 697 Warren-Girder Loaded at the Middle, and at an Intermediate Point — Uniformly Loaded — Rolling Load, 699 Parallel 1-attice-Girder. 708 Parallel Strut-Girder, 708 Roofs, 713 WORK, OR LABOUR. Units of Work or Labour: — Horse-power — Mechanical Equivalent of Heat — Labour of Men, 718 Labour of Horses — Work of Animals Carrying Loads, 720 FRICTION OF SOLID BODIES. Laws of Friction:-— Friction of Journals — Friction of Flat Surfaces, . . 722 Friction on Rails: — M. Poiree's Experiments 724 Work and Horse-power Absorbed by Friction:— Formulas, . . . 725 MILL-GEARING. Toothed Gear:— Pitch of the Teeth of Wheels— Spur Fly-wheels— Toothed Wheels for Millwork — Rules, 727 Form of the Teeth of Wheels, . . .731 Proportions of the Teeth of Wheels, 734 Transverse Strength of the Teeth of Wheels — Working Strength, . . . 735 Breadth of the Teeth of Wheels, 737 Horse-power Transmitted by Toothed Wheels, 737 Weight of Toothed Wheels, . . 739 Frictional Wheel-Gearing, 741 Belt- Pulleys and Belts.-— Tensile Strength, 742 Horse-power Transmitted by Belts, 743 Adhesion and Power of Belts — Examples of very wide Belts, .... 744 India-rubber Belting, 750 Weight of Belt- Pulleys, 750 Rope Gearing: — Transmission of Power by Ropes to Great Distances, . . 753 Cotton Ropes^ . , , . , 755 CONTENTS. XVll PAGB Shafting: — ^Transverse Deflection of Shafts, ,756 Ultimate Torsional Strength of Round Shafts, 758 Torsional Deflection of Round Shafts, 759 Power Transmitted by Shafting 760 Weight of Shafting, 761 Strength and Horse-power of Round Wrought-iron Shafting, .... 762 Frictional Resistance of Shafting, 763 Ordinary Data for the Resistance of Shafting, ... ... 763 Joomals of Shafts, 766 EVAPORATIVE PERFORMANCE OF STEAM-BOILERS. Normal Standards, 768 Heating Power of Fuels:— Table of Heating Power, 769 E%'APORATivE Performance of Stationary and Marine Steam - Boilers, WITH Coal: — Surplus Air Admitted to the Furnace, .... 770 Experiments on the Evaporative Power of British Coals, by Delab^che and Playfair, 770 Evaporative Performance of Lancashire Stationary Boilers at Wigan — With Economizer and Without Economizer ~ - Water-tubes — Temperature of the Products of Combustion, and of the Feed-water — Trials of D. K. Clark's Steam-Induction Apparatus — Of Vicars' Self- feeding Fire-grate, . 771 Evaporative Performance of South Lancashire and Cheshire Coals in a Marine Boiler, at Wigan 781 Trials of Newcastle and Welsh Coals in the Wigan Marine Boiler, . . 784 Evaporative Performance of Newcastle Coals in a Marine Boiler, at Newcastle- on-Tyne, 785 Trials of Newcastle and Welsh Coals in the Marine Boiler at Newcastle, for the Board of Admiralty, 787 Trials of Welsh and Newcastle Coals in a Marine Boiler at Keyham Factory, . 790 Evaporative Performance of American Coals in a Stationary Boiler, . . . 791 Evaporative Performance of an Experimental Marine Boiler, Navy Yard, New York, 795 Evaporative Performance of Stationary Boilers in France, .... 796 Evaporative Performance of Locomotive Boilers, 798 Evaporative Performance of Portable- Engine Boilers, 801 Relations of Grate-Area and Heating Surface to Evaporative Per- formance: — Mr. Graham's Experiments — Experiments by Messrs. Woods and Dewrance — Experimental Deductions of M. Paul Havrez, . . . 802 Formulas for the Relations of Grate-Area, Heating Surface, Water, and Fuel:— General Equations, 804 Formulas for the Experimental Boilers, 807 General Formulas for Practical Use, 819 Table of the Equivalent Weights of Best Coal and Inferior Fuels, . . . 820 STEAM-ENGINE. Action of Steam in a Single Cylinder:— The Work of Steam by Expan- sion — Clearance — Formulas for the Work of Steam — Initial Pressure in the Cylinder — Average Total Pressure in the Cylinder — Average Effective Pres- snre — Period of Admission and the Actual Ratio of Expansion — Relative Performance of Equal Weights of Steam Worked Expansively — Proportional Work Done by Admission and by Expansion — Influence of Clearance in Redndng the Performance of Steam 822 Table of Ratios of Expansion of Steam, with Relative Periods of Admission, Pressures, and Total Performance, 835 b XVlll CONTENTS. PACK Total Work Done by One Pound of Steam Expanded in a Cylinder, . 838 Consumption of Steam Worked Expansively per Horse-power of Net Work per Hour, 840 Table of the Work Done by One Pound of Steam of lOO-lbs. Pressure per Square Inch, 841 Net Cylinder-Capacity Relative, to the Steam Expanded and Work Done in One Stroke, 843 Table of ditto, 844 Compound Steam-Engine; — Woolf Engine— Receiver-Engine— Ideal Diagrams, without Clearance — Work of Steam as Affected by Intermediate Expansion — Intermediate Expansion — Work, with Clearance — Comparative Work of Steam in the Wool! Engine and the Receiver- Engine, 849 Formulas and Rules for Calculating the Expansion and the Work of Steam, . 869 Compression of Steam in the Cylinder, 878 Practice of the Expansive Working of Steam: — Actual Performance — Data — Deductions — Conclusions, 879 FLOW OF AIR AND OTHER GASES. Discharge of Air through Orifices — Anemometer, 891 Outflow of Steam through an Orifice, 893 Flow of Air through Pipes and Other Conduits 894 Resistance of Air to the Motion of Flat-Surfaces 897 Ascension of Air by Difference of Temperature, 897 WORK OF DRY AIR OR OTHER GAS, COMPRESSED OR EXPANDED. Work at Constant Temperatures:— Isothermal Compression or Expansion, 899 Work in a Non-conducting Cylinder, Adiabatically, .... 901 Efficiency of Compressed-Air Engines, 909 Compression and Expansion of Moist Air, 912 AIR MACHINERY. Machinery for Compressing Air and for Working by Compressed Air: — Compression of Air by Water at Mont Cenis Tunnel Works — By Direct-action Steam-pumps — Compressed-air Machinery at Powell Duffryn Collieries, 915 Hot- Air Engines: — Laubereau's— Rider's — Belou*s — ^Wenham's, . . . 917 Gas-Engines: — Lenoir^s — Hugon*s — Otto & Langen's, 920 Fans or Ventilators: — Common Centrifugal Fan— Mine- Ventilators — GuibaFs Fan — Cook's Ventilator, 924 Blowing Engines 926 Root's Rotary Pressure-Blowers, 92^ FLOW OF WATER. Flow of Water through Orifices: — Formulas — Mr. Bateman's Experi- ments, 929 Mr. Brownlee's Experiments with a Submerged Nozzle, 931 Flow of Water Ovi;R Waste-Boards, Weirs, &c., 932 Flow of Water in Channels, Pipes, and Rivers, 932 Cast-Iron Water Pipes, 934 C AST-Iron Gas Pipes, 936 CONTENTS. • XIX WATER-WHEELS. PAGE Wheels on a Horizontal Axis:— Undershot- Wheels— Paddle- Wheels— Breast- Whcels — Overshot- Wheels, 937 Wheels on a Vertical Axis:— Tub— Whitelaw*s Water-mill— Turbines- Tangential Wheels 939 MACHINES FOR RAISING WATER. Pumps: — Reciprocating Pumps — Centrifugal Pumps — Chain Pump — Noria, . 944,. 968 Water- works Pumping Engines, . . . . 948 Hydzanlic Rams, 949 HYDRAULIC MOTORS. Hydraulic Press, 950 Armstrong's Hydraulic Machines, 950 FRICTIONAL RESISTANCES. Steam Engines, 951 Tools: — Shearing Machines — Plate-bending Machines — Circular Saws, '951 VTork of Ordinary Cutting Tools, in Metal, 952 Screw-cutting Machines — Wood-cutting Machines — Grindstones, . 954 Colliery Winding Engines, 956 Waggons in Coal Pits, 956 Machinery of Flax Mills: — ^M. Comut*s Experiments, 957 Hoise-|x>wer Required, 959 Machinery of Woollen Mills:— Dr. Hartig's Experiments, . . . , . 959 Machinery for the Conveyance of Grain, 960 Traction on Common Roads: — M. Dupuit's Experiments — M. Debauve's De- ductions — M. Tresca's Experiments, 961 Carts and Waggons on Roads and on Fields, 962 Resistance on Railways, 965 Resistance on Street Tramways, 966 APPENDIX. Dr. Siemens' Water Pyrometer, 967 Atmospheric Hammers, 967 Bernays* Centrifugal Pumps, 968 Steam-Vacuum Pump, . 969 Index, . - .'.... 971 AUTHORITIES CONSULTED OR QUOTED. American, United, Railway Master Car- Builders* Association, Standard Sizes of Bolts and Nuts by, 663. American Society of Civil Engineers, Journal of: — Mr. J. F. Flagg, on Steam-vacuum Pamps, 969. Anderson. Dr., on the Strength of Cast Iron, 555- Anderson, William, on Heating Water by Steam, 465. 466, 468 ; Translation of Cher- noffs Paper on Steel, 616. Annales dts Mines: — M. Krest, on the Slip of Belts, 742. Annales des Fonts et Chaussies: — M. Him's Rope Transmitter of Power, 754. Annales du Ginie Civile: — M. Paul Havrez, on Heating Surface of Locomotives, 803. A mnals of Philosophy : — Mr. Dunlop, on Tor- sional Strength of Cast Iron, 565. Annuaire det Association des Inginieurs sort is de r^coU de Zi^/.— Rivelted Joints, 641. Armengaud, French Standard Bolts and Nuls, by, 683. Aimstrong. Sir Wm., on Evaporative Power of Coals, 785; his Hydraulic Machinery, 950- Arson, Anemometer by. 892. A^by & Co., Work of Steam in Portable Engine by, 883. Audenet, on Surface-Condensers, 475. B BaJcer. B., on the Strength of Beams, 512; of Oak. 544, 549; of Columns, 645, 646; of RaQs, 662, 666. Barlow. Peter, on Strength of Timber, 547 ; of Cast Iron, 561 ; of Wrought Iron, 567, 588, 590 ; of Iron Wire, 586. Bariow. W. H,. on the " Resistance of Flex- ure." 507. Bamaby. Mr., on Strength of Punched Steel Plate, 642. Barrow Hematite Steel Company, Strength of Steel made by, 594. 618, 619, 620, 621. Batcman, J. F., on Flow of Water through Submerged Openings. 930; his Cast-Iron Pipes. 934- Baudrimont, on Strength of Metallic Wires. 628. Beardmore, on the Work of Horses, 720 ; on Limits of Velocity at the Bottom of a Channel, 934. Beaufoy, Colonel, on Resistance of Air, 897. Bell, J. Lothian, on the Heat in Blast Fur- naces, 498. Berkley, George, on the Strength of Cast- iron Beams. 647-650. Berkley, J., Specific Gravity of Indian Woods, by, 209. Bemays, Joseph, on Centrifugal Pumps, 968. Bertram, W., on Rivetted Joints, 634-637. Borsig, Herr, Strength of Wrought- Iron Plates, 586. Box, Thomas, on the Load on Journals, 766 ; Thickness of Gas Pipes, by, 936. Boyden, Outflow Turbine by, 940. Bradford, W. A., on Otto and Langen's Gas- Engine. 924. Bramwell. F. J., on Heating Water by Steam, 467, 468: on the Strength of Cast Iron, 556 ; on Portable Steam Engines, 801, 883, 886 ; on the Expansive Working of Steam, 889. Brereton, R. P., on Strength of Timber Piles, 646. Briggs, Blowing Engine by, 927. British Associatiom, Transactions of : — F. W. Shields, on Strength of Cast-iron Columns. 645- Brown & May, Work of Steam in Portable Engine by, 882. Brownlee, J., on Saturated Steam, 382; on the Outflow of Steam, 893 ; Flow of Water through a Submerged Nozzle, 931. Bruce, G. B., on the Work of a Labourer, 719. Brunei, on the Strength of Rivetted Joints, 638 ; and of Bolts and Nuts, 680. Buchanan, W. M., on Saturated Steam, 379. Buckle. W., on Fans, 924. Buel, R. H., on the Slip of Belts, 742. Bulletin de la Sociiti Industrielle de Mul- house: — M. Leloutre on Steam Engines, 886. Burnat, on Condensation of Steam in Pipes, 472. 474- Bury, Wm., on Strength of Flat Stayed Sur- faces, 686. <52 XXll AUTHORITIES CONSULTED OR QUOTED. Cameron, Dr. , Analysis of Peat by, 454. Chari^-Marsaines, on Flemish Horses, 964. Chenot Ain^, Atmospheric Hammer by, 967. Chemoff, on Steel, 616. Chevandier, on Composition of Wood, 440; on its Weight and Bulk, 442, 443. Chevandier & Wenheim, on Strength of Tim- • ber, 538, 545, 546, 549. Clark, D. t^., on Proi>erties of Saturated Steam, 387; on Locomotive Boilers, 798; on the Work of Steam. 879, 880^ 884; on Resistance on Railways, 965. Clark, Edwin, on the Strength of Beams, 51a; of Red Pine, 543, 544, 549 ; of Cast Iron, 562 ; of Bar Iron, 570, 588, 590, 623. Clark, Latimer, on Wire Gauges, 130. Claudel on Fuels and Woods, by, 207, 211, 212 } tints of Heated Iron, 328 ; on Heating Factories, 486; on Heating Rooms, 488, 489; on Belts, 743, 746; on Blowing En- gines, 927; on Pumps, 944. Clement, on Transmission of Heat, 462, 468 ; on Condensation of Steam in Pipes, 474 ; on Drying Stuffs, 496 ; on the Heat to Melt Iron, 497. Cochrane, J., on Strength of Perforated Bar Iron, 633. Cockerill, John, Blowing Engines by, 927. Colliery Guardian : — Mr. Mackworth on Ven- tilation of Mines. 480. Conservatoire cUs Arts et Metiers, Annates du: — Hot-Air Engines by Laubereau. and by Belou, 917-9x9 ; Gas-Engines by Lenoir, 920; by Hugon, 921; by Otto & Langen, 923- Cooper, J. H., on Very Wide Belts, 747, 749. Comet, on the Work of a Labourer in France, 720. Comut, E., on Mill-Shafting, 766 ; on Machin- ery of Flax-Mills, 957 ; on Flow of Air in Pipes, 896. Cotterill. J. H., on Work of Compression of Air, 903. Cowper, E. A., Compound Engine by, 889. Craddock, Thomas, on Cooling through Plates, 461. Crighton & Co. , on Drying Grain, 496. Crookewitt, on Specific Gravities of Alloys, 2CO. Crossley, F. W., on Otto & Langen's Gas- Engines, 923. Cubitt, Mr., on Strength of Cast-Iron Beams, 649. D DagUsh, G. H., on Resistance of Colliery Winding Engines, 956. Dalton, Dr., on " Spontaneous " Evaporation of Water, 491. Daniel, W., on Ventilation of Mines, 925. Dan vers, F. C, on Coal Economy, 4x0. Darcy, on Cooling Hot Water in Pipes, 469. D'Aubuisson, on Flow of Compressed Air, 896 ; on Hydraulic Rams, 949. Davey, Paxman,. & Co., Work of Steam in Portable Engine by, 883. Davies. Thomas, on Strength of Rivetted Joists. 658. Davison, R., on Resistance of Shafting, 766; Duty of Pumps by, 944; on Resist- ance of Grain Machinery, 961. Day, Summers, & Co., Work of Steam in Marine Engines by, 882. Debauve, on Resistance on Common Roads, 961. Delabtehe & Playfair, on British and Foreign Coals, 206, 413, 416, 770. Despretz, on Conducting Powers of Bodies, 331- Deville, Sainte-Claire, on Composition of Petroleum and other Oils, 456, 457. Dewrance, John, on the Heating Surface of a Locomotive, 803. Donkin, Bryan, & Co., Work of Steam in Stationary Engines by, 882. Downing, on Flow of Water in Pipes, 933. 934- Dunlop, on Strength of Cast Iron, 565. Dupuit, on Resistance on Common Roads, 961. Durie, James, on Rope-Gearing, 753. Duvoir, Ren^, Drying House by, 495. Eastons & Anderson, on Portable Steam Engines, "Box ; on Rider's Hot-Air Engine, 9x7; on Resistance 'of Waggons. 962. Elder, John, & Co., on the Strength of Boilers. 638, 693 ; Work of Steam in Marine Engine by, 882. Emery, on American Marine Engines, 884. Engineer, ZA*?.— Crighton & Co. on Drying Grain, 496 ; Mr. W. S. Hall on the Strength of Rivetted Joints, 641 ; Messrs. Woods & Dewrance on Locomotive Boilers, 803 ; Mr. C. L. Hett on Hydraulic Rams, 949. Engineering: — on Heating Water by Steam, 464; on Cooling Wort, 470, 471; Mr. B. G. Nichol on Surface Condensation, 476; Mr. G. Graham Smith on Strength of Timber, 544; Factor of Safety for Wrought Iron, by Roebling, 625 ; Mr. W. S. Hall on the Strength of Rivetted Joints, 64X ; Mr. John Mason on Strength of Untanned Leather Belts, 680; Mr. Phillips on Strength of Flat Plates, 686; Mr. Bury on the Strength of Flat Stayed Surfaces, 686 ; Messrs. John Elder & Co. on the Strength of Boilers, 638, 693 ; Mr. J. Durie on Rope Gearing, ^ AUTHORITIES CONSULTED OR QUOTED. XXlli 753; Dr. Martig- on Resistance of Tools, 951; Resistance of Waggons, by Messrs. Eastons & Anderson, 962. Emglisk Mechanic : — Mr. W. A. Bradford on Olto & Langen's Gas-Engine. 924. E^Tard« A. on the Work of Animals, 720. Fagersta Steel Works, Strength of Steel made at, 604, 618, 619. 690, 621. Faiifaaim, Sir Williani, on Hot-Blast Iron, 556; on the Strength of Cast Iron, 557; on the Strength of Wrought Iron, 567-569; of Rivetted Joints, 633 ; of Screwed Stay- Bolts and Flat Stayed Plates. 685 ; on the Proportions of Spur Wheels, 729, 734, 737 ; on the Load on Journals. 766, 767; on Water Wheels, 938. Fairbaim & Tate, on the Expansion of Steam, 383- Fairweather, James C, on Resistance of Air, 897- Faraday, Dr., on the Liquefaction of Gases, 372. Favre & Silbermann, on the Heating Powers of Combustibles, 404. Field. Joshua, on the Work of Labourers, 719. Fincham, on Strength of Timber, 542, 543, 549- Flagg, J. F., on Steam-vacuum Pumps, 969. Fletcher, L. E., on the Strength of a Boiler, 638. 693 ; his Reports. 696 ; his Report on Bofler and Smoke Prevention Trials, 771- 784. Yowke, Captain, on Colonial Woods. 209. Fowler, G., on Resistance of Waggons in Coal Hts, 956. Fowler, John, Strength of Steel Rails de- signed by, 666, 670. Fowler, J., & Co., Compressed-air Machinery by, 916. Fox. Head, & Co., on Condensation of Steam in a Boiler, 475. Francis, J. B., on a Swain Turbine, 943. Franilin Institute, Journal of: — the Shear- ing Resistance of Bar Iron, by Chief Engineer W. H. Shock, 588; Mr. R. H. Bud on Belts, 742 ; Mr. H. R. Towne on Belts, 742, 745; Mr. J. H. Cooper on Bells, 747: Mr. S. Webber on Mill Shaft- ing. 763, 764; Mr. Emery on American Marine Engines, 884; Mr. Briggs on Blowing Engines, 927; Mr. J. B. Francis on a Swain Turbine, 943; Mr. E. D. Leavitt's Pumping Engines. 948. Gammelbo & Co., Hammered Bars made by, Strength of, 581. GaudiUot; on Heating Apparatus, 486. Gay-Lussac. on Cold by Evaporation, 376. Glynn, Mr.| on Strength of Ropes, 673 ; on the Work of a Labourer, 718. Gooch, Sir Daniel, on Consumption of Water by the "Great Britain" Locomotive. 884. Gordon. L. D. B., on Strength of Colimms, 645- Graham, John, on Heating Water, 461 ; on Heating Surface, 802. Grant, on Strength of Cements, &c., 630. Greaves, on Pumping Engines, 948. Grouvelle, on Condensation of Steam in Pipes, 474; on Heating Factories, 486. 487. H Hackney, W., on Anthracitic Cbke, 432. Haines, R.. on Indian Coals, 423. Hall, W. S., on the Strength of Rivetted Joints, 64X. Harcourt, Vernon, on Analysis of Coal-Gas, 458. Harmegnies, Dumont, & Co., on French Wire Ropes, 677. Hartig, Dr., on Driving Belts, 743; on Re- sistance ' of Tools, 951 ; on Resistance of Machinery of Woollen Mills, 959. Havrez, P.. on Heating Water by Steam, 464, 468; on Heating Surface of Loco- motives, 803. Hawksley, Thomas, on Flow of Air through Pipes, 894 ; on Velocity of Air in Up-cast Shaft, 897 ; on Flow of Water in Pipes. 933 ; on Thickness of Water Pipes, 935. Hett, C. L., on Hydraulic Rams, 949. Hick, John, M.P., on Friction of Leather ColUurs, 950. Him, on Work of Expanded Steam in Sta- tionary Engines, 886. Hodgkinson, on the Strength of Cast Iron, 553-555. 558, 559. 563. 564; of Columns, 643, 646; of Cast-iron Flanged Beams, 647-650. Holtzapffel, his Wire-Gauges, 131, 13a, 134. Hood, on Warming and Ventilation, 477-485. Hopkinson, on the Performance of a Corliss Engine, 88 z. Hunt, R., on Combustion of Coal, 770. Hutton, Dr., Law of Resistance of Air by« 897. JLa Institute of Naval Architects, Transactions of M^.— Strength of Rivet Joints of Sted Plates, 642. Institution of Civil Engineers, Proceedings of:— Mr. Wm. Anderson on Heating Water by Steam, 465; M. Bumat on Condensation of Steam in Pipes. 472; Dr. Pole on Spon- taneous Evaporation, 493; Regenerative Hot-Blast Stoves. 556: Mr. Bramwell on XXIV AUTHORITIES CONSULTED OR QUOTED. Strength of Cast Iron, 556; Mr. Grant on | the Strength of Cements, &c., 630; Mr. J. Cochrane on the Strength of Punched Bar Iron, 633; Mr. R. Price Williams on Strength of Rails, 662; Mr. J. T. Smith on the Strength of Bessemer Steel Rails, 664; Mr. R. Davison on Resistance of Shafting, 766 ; Evaporative Performance of Steam Boilers in France, 796 ; Composition of Coals and Lignites, 797; M. Paul Havrez on Heating Surface of Locomotives, 803; Mr. Emery on American Marine Engines, 884; Mr. Hawksley on Flow of Air through Pipes, 894 ; and on Velocity of Air in Up- cast Shaft, 897 ; M. Piccard on the Work of Compressed Air, 911 ; Mr. J. B. Francis' trial of a Swain Turbine, 943; Mr. R. Davison on Duty of Pumps, 944 ; Hon. R. C. Parsons on Centrifugal Pumps, 947 ; Mr. Henry Robinson on Armstrong's Hydraulic Machines, 950. Institution of Engineers and Ship-Buiiders in Scotland, Transactions of the: — on Strength of Helical Springs. 672 ; Report on Safety Valves, 893; Mr. J. Brownlee's Experi- ments on Flow of Water, 931. Institution of Mechanical Engineers, Pro- ceedings of: — Mr. C. Little on the Shearing and Punching Strength of Wrought Iron, 587 ; Mr. Vickers on the Strength of Steel, 621; Mr. W. R. Browne's paper on Rivetted Joints, 637; Mr. Robertson on Grooved Frictional Gearing, 741 ; Mr. H. M. Mor- rison on Him's Rope Transmitter, 755; Mr. Ramsbottom on Cotton- Rope Trans- mitter, 755; Mr. Westmacolt and Mr. B. Walker on Resistance of Shafting, 766 ; Mr. D. K. Clark on the Expansive Working of Steam in Locomotives, 879. 880; Data of the Practical Performance of Steam, 880; Mr. F. J. Bramwell on Economy of Fuel in Steam Navigation, 889 ; Compressed-Air Machinery by Messrs. John Fowler & Co., 916 ; Wenham's Hot-Air Engine, 919 ; Mr, F. W. Crossley on Otto and Langen's Gas- Engine, 923; Mr. Buckle on Fans, 924; Mr.J.S.E. Swindell on Ventilation of Mines, 925; Mr. W. Danielon Ventilation of Mines, 925 ; Mr. A. C. Hill on Blowing Engines, 927 ; Mr. J. F. Bateman's Experiments on Flow of Water, 930 ; Mr. David Thomson on Pumping Engines, 948; Mr. G. H. Daglish on Winding Engines. 956 ; Mr. G. Fowler on Resistance of Waggons in Coal Pits, 956 ; Mr. Westmacott on Com- Ware- housing Machinery, 961. Iron and Steel Institute, Journal of the: — Mr. J. Lothian Bell on the Cleveland Blast Furnaces, 498. Isherwood, Trials of Evaporative Performance of a Marine Boiler, 795. J James, Captain, on the Strength of Cast Iron, 555- Jardine, Mr., on the Strength of Lead Pipes, 696. Johnson, Professor W. R., on American Coals, 418, 770. 791-795- Joule, Dr., Mechanical Equivalent of Heat, by, 332. K Kane, Sir Robert, on Peat, 453. Kennedy, Colonel J. P., on Weight and Volume of Goods carried on Railways, 213. Kirkaldy, David, on Compressive Strength of Timber, 546, 547, 647 ; on the Tensile Strength of Wrought Iron and Steel, 571- 578 ; of Swedish Hammered Bars. 581, 590; of Krupp and of Yorkshire Iron Plates, 583-586 ; of Borsig's Iron Plates, 586; Ten- sile Strength of Bar Steel 593, 594 ; of He- matite Steel, 594 ; of Krupp Steel, 595 ; of Steel Bars, for the Steel Committee, 597- 600; of Fagersta Steel, 604-611 ; of Siemens- Steel Plates and Tyres, 612-614 ; on Shear- ing Strength of Steel. 617; on Strength of Phosphor-Bronre, 628, 629; of Wires, 629; of Rolled Wrought-iron Joists, 654; of Rails. 662, 663, 666-668; of Ropes, 674; of Belt- ing, 680; of Plates of a Marine Boiler, 694. Krest, on the Slip of Belts, 742. Krupp, Herr, Strength of Wrought-Iron Plates made by, 583 ; of his Cast Steel, 595. 618- 62T. L Landore Siemens-Steel Company, Strength of Steel Plates and Tyres made by, 6i*-6i4. Laslett. Thomas, on the Strength of Timber, 538-542. 546, 548, 550. 647. Leavitt, E. D., Pumping Engines by. 948. Legrand, on Boiling Points, 370. I^igh, Evan, on Belting, 746. Ldoutre, on M. Him's Experiments on Work of Steam, 886. Leplay. on Moisture in Wood, 439; on Drying Wood, 496. Literary and Philosophical Society of Man- chester, Memoirs of: — Dr. Dalton on " Sipontaneous " Evaporation, 491; Mr. John Graham on Heating Surface, 802. Little, C, on the Shearing and Punching Strength of Wrought Iron, 587. Lloyd, Thomas, on the Strength of Bar Iron. 569. 570. London Association of Foremen Engineers, Proceedings of: — Mr. David Thomson on Expansive Work of Steam, 822. Longridge, J. A., on Combustion and Evapn orative Power of Coals, 770, 785. Longsdon, Mr., on Strength of Krupp Steel, 595- AUTHORITIES CONSULTED OR QUOTED. XXV M MacCon, on the Strength of Rivetted Joints, 641. Mackintosh. Charles, Weight of Belt-Pulleys by. 75a. Mackworth. H., on Ventilation of Mines, 479- Madure. H. H., on Strength of Timber, 542, 543. 549- Macneil, Sir John, on Resistance on Common Roads; 964. Mahan, Lieutenant F. A., on Outward-Flow Tuibines, 941. Mallaid, on Compressed-Air Machines, 90a ; on Compressed Air, 907, 91a. Mallet, R.. Strength of Buckled Iron Plates by, 660. Marshall Sons, & Co., Work of Steam in Portable Engine by, 883. Mason, John, Strength of Untanned I.^eather Belts by, 680. M'Donndl, A, on Composition of Peat, 454. Menelaus, on Portable Steam Engines, 801. Miller, T. W., Trials of Coals by, 790. Miner & Taplin. Trials of Coals by, 787. Montgoilfier, on Drying by Forced Currents, 494- Monthly Reports to the Manchester Steam- Users' Association: — Mr. L. E. Fletcher's Data. 696. Morin. on Transverse Strength of Timber, 537; on the Friction of Journals, 722 ; and 1 of Solid Bodies, 733 ; on Leather Belts, 743-745; *5^ Breast Wheels, 938; on a Foomeyron Turbine. 940; on Centrifugal Pomps, 946. Morrison. H. M., on M. Him's Rope Trans- mitter, 755- Morton, Francis, & Co., Weight of Jron fleets by, 245 ; Strength of Cable Fencing Stands by, 676. Moser. Strength of Beams tested for, 654. Muspratt. Dr.. Analyses of Coke by, 433. N Naper, James R., on Transmission of Heat, 460 ; on Drying Stuffs, 496. Nao. on Moisture in Charcoal, 451. Xen-all. R. S., & Co., Strength of Hemp and Wire Ropes by, 674. Nichol, B. G., on Condensation of Steam in Pipes and Tubes, 474, 476. Xjcon & Lynn. Trials of Coals by, 784. Norris & Co.. Strength of Leather Belts by, 68a North British Rubber Company, Driving Belts by. 730- Af'orth of England Mining Institute, Transac- tions e^.— Rivetted Joints, 588. O Oldham, Dr., on Indian Coals, 424. Ott, Karl Von, on Strength of Ropes, 674, 679. P Parsons, on Strength of Oak Trenails, 551. Parsons, Hon. R. C, on Centrifugal Pumps, 947- Payen. on Explosive Mixture of Gas and Air, 921. Pearce, W. A., on Rope Gearing, 754. Peclet, on Radiation of Heat, 329 ; on French Coals. 420 ; on Coke, 431 ; on Moisture in Tan, 455; on Transmission of Heat, 459, 46a, 463, 468; on Condensing Power of Air and Water, 475 ; on Ventilation, 477 ; on Heating Apparatus, 488, 489 ; on Drying by Air Currents, 494 ; on a Drying House, 495 ; on Cupola Furnaces, 497. Penot, on Drying Houses, 496. Penrose & Richards, their Anthracitic Coke, 432- Perkins, Heating Apparatus by, 486. Perkins, Jacob, Invention of the Ice-Making Machine by, 373. Person, on the Latent Heat of Fusion, 367. Phillips, on Strength of Flat Plates, 686. Piccard, on Work of Compressed Air, 911. Poir^ on Friction on Rails by, 724. Pole, Dr., on Spontaneous Evaporation, 493 ; on the Strength of Steel Wire, 617. Poncelet, on Water Wheels, 938. Forte/euille de John Cockerill: — Blowing Engines, 927. Porter, C. T., on Expansion of Steam, 886. Pouillet, on Luminosity at High Temper- atures, 328. R Radford, R Heber, Weight of Belt-Pulleys by,. 751. 752. Ramsbottom, J., on Cotton- Rope Transmitter, 755- Rankine, Dr. , on Expansion of Water, 340 ; on the Melting Point of Ice. 364; on Transmission of Heat, 461 ; on Shearing Strength of Oak Trenails, 551 ; and of Cast Iron, 561 ; Factors of Safety, 625, 626 ; on Stresses in Roofs, 715, 717 ; on Load on Working Surfaces. 767. Reading Engine Works Co., Work of Steam in Portable Engine by, 883. R^clus, Specific Gravity of Sea Water by, 126. Regnault, Air Thermometer by, 325 ; on the Expansion of Air, 344 ; on Specific Heat of Metals, 353; and Gases, 359; Boiling Points of Vapours, 371 ; on Steam, 378, 379, 383, 384 ; on the Mixture of Gases and XXVI AUTHORITIES CONSULTED OR QUOTED. Vapours, 392 ; on French Coals, 420, 421 ; on Lignite and Asphalte, 436. Reilly, Calcott, on the Varieties of Stress, 500- Rennie, on the Work of Horses, 720: Reime IndustrielU: — Atmospheric Hammer by M. Chenot Ain^, 967. Reynolds, Dr., on Peat, 454. Richardson, Dr., on Coals, 412 ; on Coke, 433; Report on Evaporative Power of Coals, 785- Robertson, James, on Grooved Frictional Gearing, 741. Robinson, Henry, on Armstrong's Hydraulic Machines, 950. Roebling, on the Strength of Iron Wire, 587 ; and of Steel Wire, 617 ; Factor of Safety for Iron, 625 ; on the Strength of Wire Rope and Hemp Rope, 676. Ross, Owen C. D., on Coal Gas, 457. Rouget de Lisle, on Drying StufTs, 496. Royal Society of Edinburgh, Proceedings of: — Mr. Fairweather on Resistance of Air, 897. Royer, on Drying Houses, 496; on Drying Stuffs, 496. Russell & Sons, J., on the Strength of Wrought-Iron Tubes, 692, 693. Ryland Brothers, Warrington Wire Gauge by, 133. 247- Sauvage, on Charcoal, 447, 449, 452. Scheurer-Kestner & Meunier - DoUfus, on French and other Coals, and Lignites, 422, 797- Sharp, Henry, on Rivetted Joints of Steel Plates, 642. Shields, F. W., on Cast-Iron Columns, 645. Shock. Chief Engineer W. H., on Shearing Strength of Bar Iron, 587. Siemens, Dr. C. W., on Isolated Steam, 383; on the Consumption of Fud in Metallurgical Furnaces, 497; on the Strength of Hot- Blast Iron, 556 ; on Hot-Air Engines, 920 ; his Water Pyrometer, 967. Simms, F. W., on the Work of Horses, 720. Smeaton. on the Power of Labourers. 718. Smith, G. Graham, on Strength of Timber, 543. 544. 549- Smith, J. T., on Punching Resistance of Steel, 617 ; on the Strength of Rails, 664. Snelus, G. J., Analysis of Welsh Coal by, 413. SocUU IndustrielU de Mulhouse: — on Steam Boilers, 796. Soctiti IndustrielU Minerale, Bulletin de la: — M. Comut on Compressed-Aif Machi- nery, 896 ; M. Mallard on Compressed-Air Machines, 902. Sociiti des InginUurs Civils, Comptes Rendus de la: — Anemometer by M. Arson. 892. SociH6 Vaudoise des Inginieurs et des Archi- tecies, Bulletin de la: — M. Piccard on Compressed Air, 911. Society of Arts, Committee of, on Resistance on Common Roads, 963. Society of Arts, Journal of: — on Resistance on Common Roads, 963. Spill, Strength of Belting by, 680. Steel Committee of Civil Engineers, on the Strength of Wrought Iron, 579, 580 ; and of Steel, 596-603, Stephenson, Robert, on the Strength of Cast Iron, 555, 561. Stoney, on Stress in a Curved Flange, 525; on Sectional Area of a Continuous Web, 526; on Shearing Strength of Cast Iron, 561 ; his Factors of Safety, 625 ; on the Re- sistance of Columns, 643, 645, 646; on Stresses in Roofs, 715. Sullivan, Dr., on Peat, 207. SutclifTe, on Condensation of Steam in the Cylinder, 880. Swindell, J. S. E,, on Ventilation of Mines, 935- Sylvester, Cockle Stove by, 488. Tangye, J., on the Compressive Resistance of Wrought Iron, 582. Tasker, Work of Steam in Portable Engine by, 883. Telford, Thomas, on the Strength of Wrought Iron, 567 ; and of Iron Wire, 586. Thomas & Laurens, on Brown Charcoal, 449; on Heating by Steam, 463, 468. Thomson, David, on Expansive Action of Steam, 822, 882; on Centrifugal Pumps, 946 ; Duty of Pumping Engines, 948. Thomson, Professor James, Vortex Wheel by. 943- Thurston, on the Strength of Iron Wire, 587. Thwaites & Carbutt, on Root's Blower, 928. Towne, H. R., on Leather Belts, 679, 742, 745. 748-750- Tredgold, Weight and Volume of Various Substances by, 213 ; on Cooling Hot Water, 469; on Cooling of Steam in Pipes, 472, 474 ; on the Work of a Horse, 720. Tresca, on Laubereau's Hot-Air Engine, 917; on Gas-Engines, 920. 921, 923; on Pumps, 945, 946 ; on Resistance of Tram- way Omnibus, 961. Turner, Work of Steam in Portable Engine by, 883. Tweddell, R. H., on Shafting, 763. U Umber, on M. Him's Wire Ropes. 754. Unwin, on Strength of Columns, 645. Ure, Specific Gravity of Alloys by, 20a AUTHORITIES CONSULTED OR QUOTED. XXVll Vidcers, T. K., on the Strength of Steel, 621, 622. \'»lette. on Wood, 439, 441, 44a, 445; on Charcoal, 446-448. 450, 451 W Wade, Major, on the Strength of Cast Iron, 557- Walker, B., on Resistance of Shafting, 766. Walker. John, on the Work of Labourers, 718. Webb, F. "NV.. on the Strength of Steel, 614, 621. Vfdtibcr, S.. on Mill Shafting. 763. 764, 766. Westmacott, Percy, on Shafting, 766; on Com - Warehousing Machinery, 961; on Axuibt i o og's Hydraulic Machines, 950. WhiteUw, James, Water Mill by, 939. Whitworth, Sir Joseph, Standard Wife-Gauge by, 133, 134; Strength of his Fluid-Com- pressed Steel and of Iron, 614, 615; on Resistance of Steel and Iron to Explosive Force, 622; his System of Standard Sizes of Bolts and Nuts, 681 ; Standard Pitches of Screwed-Iron Piping, 683. Wiesbach, Coefl&cients for Flow of Water, 892. Williams, R; Price, on the Transverse Strength of Rails, 662, 664. Williams, Foster, & Co., Weight of Sheet Copper by, 261. Wilson, A., on the Work of Bullocks, 720. Wilson, R., on Sttength of Perforated Iron Plates, 633. Wilson, Robert (Patricroft), on Teeth of Wheels, 732. Wood, J. & E., Work of Steam in Stationary Engine by, 882. Woods, E., and J. Dewrance, on the Efficiency of Heating Surface of a Locomotive, 803. Wright, J. G., on.Rivetted Joints, 637. A MANUAL OF RULES, TABLES, AND DATA FOR MECHANICAL ENGINEERS. GEOMETRICAL PROBLEMS. PROBLEMS ON STRAIGHT LINES. Problem I. — To bisect a straight iine^ or an arc of a circle^ Fig. i. — From the ends a, b, as centres, de- scribe arcs intersecting at c and d, and draw c d, which bisects the line, or the arc, at the point e or f. Problem IL — To draw a perpen- dicuiar to a straight line^ or a radial line to a circtilar arCy Fig. i. — Operate :c ^ .4. :b : :d *• F^5. I. — Probs. L and II. as in the foregoing problem. The line CD is perpendicular to a b : the line c D is also radial to the arc a b. Problem IIL — To draw a perpen- dicular to a straight line, from a given point in that iinc,Yig. 2. — With any radius, from the given point a, in the line B c, cut the line at b and c; with a longer radius describe arcs from b Fig. 2.— Prob. III. and c, cutting each other at d, and draw the perpendicular d a. 2d Method, Fig. 3. — ^rom any cen- tre F, above bc, describe a circle passing through the given point a, Fig. 3.— Prob. III. 3d method. and cutting the given line at d; draw D F, and produce it to cut the circle at e; and draw the perpendicular a e. GEOMETRICAL PROBLEMS 3// Method^ Fig. 4. — From a de- scribe an arc eg, and from e, with the same radius, the arc a c, cutting / '/"\ Fig. 4, — Prob. III. 3d method. the Other at c ; through c draw a Une E c D, and set off c d equal to c e ; and through d draw the perpendicu- lar AD. 4//z Method, Fig. 5. — From the given point A set off a distance a e ••c m ^' » •• .. 4 E' Fig. 5.— Prob. III. 4th method. equal to three pjrrts, by any scale; and on the centres a and e, With radii of four and five parts respec- tively, describe arcs intersecting at c. Draw the perpendicular a c. Note, — This method is most useful on very large scales, where straight edges are inapplicable. Any multi- ples of the numbers 3, 4, 5 may be taken with the same effect, as 6, 8, 10, or 9, 12, 15. Problem IV. — To draw a perpen- diadar to a straigJit line from any point without it, Fig. 6. — From the point A, with a sufficient radius, cut the given line at rand g; and frt)m these points describe arcs cutting at e. Draw the perpendicular a e. Note. — If there be no room below the line, the intersection may be taken above the line; that is to say, be- tween the line and the given point. Fig. 6.-Prob. IV. 2d Method, Fig. 7. — From any two points B, c, at some distance apart, F / B Fig. 7.— Prob. IV. ad method. in the given line, and with the radii B A, c A, respectively, describe arcs cutting at a d. Draw the perpendi- cular A D. Problem V. — 72? draw a straight line parallel to a giveti line, at a giten distance apart. Fig. 8. — From the cen- c D ,.- ^- -... 1- .... i • B Fig. 8.— Prob. V. tres A, B, in the given line, with the given distance as radius, describe arcs c, D ; and draw the parallel line c D touching the arcs. Problem VI. — To draw a parallel through a given point. Fig. 9. — With a radius equal to the distance Of the ON STRAIGHT LINES. given point c from the given line A B, describe the arc d from b, taken 1 .-^ ■--. A rig. 9-- B -Prob. VI. considerably distant from c Draw the parallel through c to touch the arc D. 2d Method^ Fig. lo. — From a, the — !p • I _t Fig. lo.— Prob. VI. ad method. given point, describe the arc f d, cut- ting the given line at f; from f, with the same radius, describe the arc e a, and set oflf f d equal to e a Draw the parallel through the points a, d. Note, Fig. II. — When a series of parallels are required perpendicular to a base line a b, they may be drawn, as in Fig. i, through points in the base line, set oflf at the required dis- t \ * •B Fig. XX.— Prob. VI. tamces apart This method is con- venient also where a succession of parallels are required to a given line, c D ; for the perpendicular a b may be drawn to it, and any number of par- allels may be drawn upon the per- pendicular. Problem VII. — To divide a straight line into a number of equal parts ^ Fig. 12. — To divide the line a b into, say, five parts. From a and b draw par- allels A c, B D, on opposite sides. Set off any convenient distance four times >-' A^ X — % X -V ^ Fig. 13.— Prob. VII. (one less than the given number) from A on AC, and from b on bd; join the first on a c to the fourth on B D, and so on. The lines so drawn divide a b as required. 2d Method^ Fig. 13. — Draw the line A c at an angle from a, set off, say. h % 9 A Fig. 13.— Prob. VII. ad method. five equal parts; draw b 5, and draw parallels to it from the other points of division in a c. These parallels divide a b as required. Note. — By a similar process a line may be divided into a number of unequal parts; setting oflf divisions on A c, proportional by a scale to the required divisions, and drawing par- allels cutting A 6. Problem VIII. — Upon a straight GEOMETRICAL PROBLEMS line to draw an angle equal to a given angle. Fig. 14. — I^t a be the given angle, and fg the line. -With any radius, from the points a and f, de- scribe arcs D E, I H, cutting the sides of the angle a, and the line f g. Set Fig. 14.— Prob. VIII. ofl the arc i h equal to d e, and draw F H. The angle f is equal to a, as required. To draw angles of 60** and 30°, Fig. 1 5. — From F, with any radius f i, de- scribe an arc i h ; and from i, with the same radius, cut the arc at h, and .H^. F K I Fig. 15.— Prob. VIII. draw F H to form the required angle I F H. Draw the perpendicular h k to the base line, to form the angle of 30*" F H K. To draw an angle of 45**, Fig. 16. — Set off the distance f i, draw the i.r I Fig. 16.— Prob. VIII. perpendicular i h equal to i f, and join H F, to form the angle at f as re- quired. The angle at h is also 45**. Problem IX. — To bisect an angle, Fig. 17. — Let A c B be the angle; on the centre c cut the sides at a, b. On A and B, as centres, describe arcs cutting at d. Draw c d, dividing the angle into two equal parts. Fig. 17. — Prob. IX. Problem X. — To bisect the inclina- tion of two lines, of which the intersec- tion is ifiaccessible, Fig. 18. — Upon the B Fig. 18.— Prob. X. given lines cb, ch, at any points, draw perpendiculars e ^, g h, of equal lengths, and through f and g draw parallels to the respective lines, cut- ting at s; bisect the angle fsg, so formed, by the line s d, which divides equally the inclination of the given lines. ON STRAIGHT LINES AND CIRCLES. 5 PROBLEMS ON STRAIGHT LINES AND CIRCLES. Problem XI. — Through tuto given points to describe an arc of a circle with a given radius^ Fig. 19. — On the points Problem XIIL — To describe a cir- de passing through three given points^ Fig. 21. — Let A, B, c be the given points, and proceed as in last pro- Fig. 19.— Proh. XI. A and B as centres, with the given radios, describe arcs cutting at c; and from c, with the same radius, describe an arc a b as required. Problem XII. — To find the centre of a cirde^ or of an arc of a drcle, I St, for a circle, Fig. 20. — Draw the Fig. 21.— Prob. XII. XIII. blem to find the centre o, from which the circle may be described. Note, — ^This problem is variously useful: — in striking out the circular arches of bridges upon centerings, when the span and rise are given; describing shallow pans, or dished iM Fig. aa— Prob. XII. chord A b, bisect it by the perpendi- cular c D, bounded both ways by the circle; and bisect c d for the centre g. 2d, for a circle or an arc, Fig. 21. — ^Select three points, a, b, c, in the circumference, well apart; with the same radius, describe 2C£t!& from these three points, cutting each other; and draw the two lines, d e, f g, through their intersections, according to Fig. i. The point o, where they cut, is the centre of the circle or arc. Fig. 82.— Prob. XIV. xst method. covers of vessels ; or finding the dia- meter of a fly-wheel or any other object of large diameter, when only a part of the circumference is ac- cessible. Problem XIV. — To describe a drcle passing through three given points when the centre is not available, 1st Mdhod, Fig. 22. — From the extreme points a, b, as centres, de- scribe arcs AH, BG. Through the third point c, draw a e, b f, cutting GEOMETRICAL PROBLEMS the arcs. Divide a f and b e into any number of equal parts, and set off a series of equal parts of the same length on the upper portions of the arcs beyond the points e, f. Draw straight lines, b l, b m, &c., to the divi- sions in A f; and a i, a k, &c., to the divisions in e g ; the successive inter- sections N, o, &c., of these lines, are points in the circle required, between the given points a and c, which may be filled in accordingly: similarly the remaining part of the curve b c may be described. 2d Methody Fig. 23. — Let a, d,b be the given points. Draw a b, a d, d b, Fig. 23. — Prob. XIV. 2d method. and ef parallel to a b. Divide d a into a number of equal parts at i, 2, 3, &c., and from d describe arcs through these points to meet ef. Divide the arc A e into the same number of equal parts, and draw straight lines from d to the points of division. The inter- sections of these lines successively with the arcs i, 2, 3, &c., are points in the circle which may be filled in as before. Note. — ^The second method is not perfectly exact, but is sufficiently near to exactness for arcs less than one- fourth of a circle. When the middle point is equally distant fi-om the ex- tremes, the vertical c d is the rise of the arc; and this problem is service- able for setting circular arcs of large radius, as for bridges of very great Fig. 24.— Prob. XV. span, when the centre is unavailable; and for the outlines of bridge-beams, and of beams and connecting-rods of steam-engines, and the like. Problem XV. — To draw a tangent to a circle from a given point in the circumference^ Fig. 24. — Through the given point a, draw the radial line — E Fig. 25. — Prob. XV. 2d method. A c, and the perpendicular f g is the tangent 2d Method^ when the centre is not available, Fig. 25. — From a, set off equal segments a b, a d; join b d, and draw A e parallel to it for the tangent Problem XVI. — To draw tangents to a circle from a point without it. Fig. 36.— Prob. XVI. ist method. \5t Method^ Fig. 26. — Draw ac from the given point a to the centre ON STRAIGHT LINES AND CIRCLES. G; bisect it at d, and from the centre D, describe an arc through c, cutting the circle at e, f. Then a e, a f, are tangents. id Method^ Fig. 27. — ^From a, with the radius a c, describe an arc b c d, and from c, with a radius equal to the •.\i> Fig. vj. — Prob. XVI. 2d method. diameter of the circle, cut the arc at B, D ; join B c, CD, cutting the circle at E,F, and draw ae, af, the tan- gents. Note. — ^When a tangent is already drawn, the exact point of contact may be found by drawing a perpen- dicular to it from the centre. Problem XVIL — Between two in- clined lines to draw a series of circles touching these lines and touching ecuh ather^ Fig. 28. — Bisect the inclination Fig. a8.— Prob. XVII. of the given lines a b, c d by the line NO. From a point p in this line, draw the perpendicular p b to the line A B, and on p describe the circle b d touching the lines and cutting the centre line at e. From e draw e f perpendicular to the centre line, cut- ting A B at F, and from f describe an arc E G, cutting a b at g. Draw g h parallel to b p, giving h, the centre of the next circle, to be described with the radius h e, and so on for the next circle i n. Inversely, the largest circle may be described first, and the smaller ones in succession. Note, — This problem is of frequent use in scroll work. Problem y>N\\l,^^ Between two inclined lines to draw a circular seg- ment to fill the angle, and touching the lines, Fig. 29. — Bisect the inclination Fig. 99. -Prob. XVIII. of the lines a b, d e by the line f c, and draw the perpendicular a f d to define the limit within which the cir- cle is to be drawn. Bisect the angles a and D by lines cutting at c, and from c, with radius c f, draw the arc H F G as required. Problem XIX. — To describe a cir- cular arc joining two circles, and to touch one of tliem at a given point. Fig. 30. — ^To join the circles a b, f g, by an arc touching one of them at f, draw the radius e f, and produce it both ways; set off fh equal to the radius ac of the other circle, join ch 8 GEOMETRICAL PROBLEMS and bisect it with the perpendicular L I, cutting E F at i. On the centre i, Fig. 30. — Prob. XIX. With radius i f, describe the arc f a as required. PROBLEMS ON CIRCLES AND RECTILINEAL FIGURES. Problem XX. — To construct a tri- angle on a given base, the sides being given, I St. An equilateral triangle, Fig. 31. Fig. 31.— Prob. XX. — On the ends of the given base, a, b, with A B as radius, describe arcs cut- ting at c, and draw a c, c b. 2d. A triangle of unequal sides, Fig. 32. — On either end of the base A D, with the side b as radius, describe an arc ; and with the side c as radius, on the other end of the base as a centre, cut the arc at e. Join a e, d e. Note, — This construction may be used for finding the position of a point, c or e, at given distances from the ends of a base, not necessarily to form a triangle. A Fig. 32.-Prob. XX. Problem XXI. — To construct a square or a rectangle onagiveti straight line, I St. A square. Fig. 33. — On the Fig. 33. — Prob. XXI. ends A, b, as centres, with the line a b as radius, describe arcs cutting at c; on c, describe arcs cutting the others at D E ; and on d and e, cut these at F G. Draw A F, B G, and join the in- tersections H, L 2d. A rectangle. Fig. 34. — On the base E F, draw the perpendiculars e h, K Fig. 34.-Prob. XXL F G, equal to the height of the rect- angle, and join g h. When the centre lines, a b, c d, Fig. 35, of a square or a rectangle are given, cutting at e. — Set off e f, eg, ON CIRCLES AND RECTILINEAL FIGURES. the half lengths of the figure, and e h, ET, the half heights. On the centres H, T, with a radius of half the length, •B 1^ A-:?' B Fig- 35.— Prob. XXI. describe arcs; and, on the centres F, G, with a radius of half the height, cut these arcs at k, l, m, n. Join these intersections. Problem XXII. — 7b construct a parallelogram^ of which the sides and one of the angles are given, Fig. 36. — CiLLl B Fig. 36.— Prob. XXII. Ehaw the side d e equal to the given length A, and set off the other side D F equal to the other length b, form- ing the given angle c. From e, with D F as radius, describe an arc, and from F, with the radius d e, cut the arc at G. Draw f o, eg. Or, the remaining sides may be drawn as |)arallels to d e, d f. The formation of the angle d is readily done as indicated, by taking the straight length of the arc h i and CI as radius, and finding the inter- section L. Problem XXIII. — To describe a circle about a triangle, Fig. 37. — Bisect two sides a b, a c of the triangle at E, F, and from these points draw per- pendiculars cutting at k. On the centre k, with the radius K a, draw the circle a b c. Fig. 37.— Prob. XXI 1 1. Problem XXIV. — To inscribe a circle in a triangle. Fig. 38. — Bisect two of the angles a, c, of the triangle by lines cutting at d; from d draw a perpendicular d e to any side, and with D E as radius describe a circle. When the triangle is equilateral, the centre of the circle may be found by bisecting two of the sides, and Fig. 38.— Prob. XXIV. drawing perpendiculars as in the pre- vious problem. Or, draw a perpen- dicular from one of the angles to the opposite side, and from the side set off one-third of the perpendicular. Fig. 39.— Prob. XXV. Problem XXV. — To describe a circle about a square, and to inscribe a square in a circle, Fig. 39. 10 GEOMETRICAL PROBLEMS I St. To describe the circle. Draw the diagonals a b, c d of the square, cutting at e; on the centre e, with the radius e a, describe the circle. 2d. To inscribe the square. — Draw the two diameters a b, c d at right angles, and join the points a, b, c, d to form the square. Note, — In the same way a circle may be described about a rectangle. Problem XXVL — To inscribe a circle in a square^ and to describe a square about a circle^ Fig. 40. ist. To inscribe the circle. — Draw Fig. 4o.-^Prob. XXVI. the diagonals a b, c d of the square, cutting at e; draw the perpendicular e F to one side, and with the radius E F describe the circle. 2d. To describe the square. — Draw- two diameters a b, c d at right angles, and produce them; bisect the angle D E b at the centre by the diameter F G, and through f and g draw per- Fig. 41.— Prob. XXVII. pendiculars ac, bd, and join the points A D and b c, where they cut the diagonals, to complete the square. Problem XXVIL — To inscribe a pentagon in a circle^ Fig. 41. — Draw two diameters a c, b d at right angles, cutting at o; bisect a o at e, and from E, with radius e b, cut a c at f ; from b, with radius b f, cut the circumference at G, H, and with the same radius step round the circle to i and k; join the points so found to form the pentagon. Problem XXVIIL — To construct a hexagon upon a given straight line^ Fig. 42. — From a and b, the ends of the given line, describe arcs cutting at^; from^, with tlie radius^ a, de- ---^ B Fig. 4a.-Prob. XXVIII. scribe a circle; with the same radius set off the arcs a g, g f, and b d, d e. Join the points so found to form the hexagon. Problem XXIX. -^72? inscribe a hexagon in a circle^ Fig. 43. — Draw a diameter AC B; from a and b as centres, with the radius of the circle a c, cut the circumference at d, e, f, g; and draw AD, D E, &c. to form the hexagon. o -^9 Fig. 43.-Prob. XXIX The points d, e, &c., may also be found by stepping the radius six times round the circle. "N ON CIRCLES AND RECTILINEAL FIGURES. II Problem XXX. — To describe a hex- a^m aJbaut a circle^ Fig. 44. — Draw a a *— . F7g. 44. — Prob. XXX. diameter adb, and with the radius A D, on the centre a, cut the circum- ference at c; join ac, and bisect it with the radius d e; through £ draw the parallel f g cutting the diameter at F, and with the radius d f describe the circle f h. Within this circle de- scribe a hexagon by the preceding problem; it touches the given circle. Problem XXXI, — To describe an octagon on a given straight line^ Fig. 45. A. Fig. 45 B -Prob. XXXI. — Produce the given line ab both ways, and draw perpendiculars ae, BF; bisect the external angles a and B, by the lines ah, b c, which make equal to a b. Draw c D and H g par- allel to A E, and equal to a b ; from the centres g, d, with the radius a b, cut the perpendiculars at e, f, and draw E F to complete the octagon. Problem XXXIL — To convert a square into an octagon. Fig. 46. — Draw the diagonals of the square cutting at e-j from the comers a, b, c, d, with a e as radius, describe arcs cutting the Fig. 46.-Prob. XXXII. sides at g, h, &c.; and join the points so found to form the octagon. Problem XXXIIL — To inscribe an octagon in a circle, Fig. 47. — Draw B Fig. 47.-Prob. XXXIII. two diameters a c, b d at right angles ; bisect the arcs ab, bc, &c., at ^,/, &c., and join a ^, ^b, &c., to form the octagon. Problem XXXIV. — To describe an octagon about a circle, Fig. 48. — Fig. 48. -Prob. XXXIV. Describe a square about the given circle .A B, draw perpendiculars hk, 12 GEOMETRICAL PROBLEMS &c., to the diagonals, touching the circle, to form die octagon. Or, the points //, k, &c, may be found by cutting the sides from the comers of the square, as in the second last problem. Problem XXXV. — To describe a polygon of any number of sides upon a given straight Hne, Fig. 49. — Produce Fig. 49. — Prob. XXXV. the given line a b, and on a, with the radius a b, describe a semicircle, divide the semi-circumference into as many equal parts as there are to be sides in the polygon; say, in this ex- ample, five sides. Draw lines from A through the divisional points d, b, and r, omitting one point a, and on the centres b, d, with the radius a b, cut A ^ at E and a r at f. Draw D E, E F, F B to complete the polygon. Problem XXXVI. — To inscribe a circle within a polygon, Figs. 50, 51. — When the polygon has an even num- ber of sides, Fig. 50, bisect two op- Fig. sa— Prob. XXXVI. XXXVII. posite sides at a and b, draw a b, and bisect it at c by a diagonal d e; and with the radius ca describe the- circle. When the number of sides is odd, Fig. 51, bisect two of the sides at a Fig. 51.— Prob. XXXVI. XXXVII. and B, and draw lines a e, b d to the opposite angles, intersecting at c; from c, with the radius c a, describe the circle. Problem XXXVI I. — To describe a circle without a polygon. Figs. 50, 51. — Find the centre c as before, and \^dth the radius c d describe the circle. The foregoing selection of prob- lems on regular figures are the most usefiil in mechanical practice on that subject Several other regular figures may be constructed from them by bisection of the arcs of the circum- scribing circles. In this way a de- cagon, or ten-sided polygon, may be formed from the pentagon, as shown by the bisection of the arc b h at ^ in Fig. 41. Inversely, an equilateral triangle may be inscribed by joining the alternate points of division found for a hexagon. Problem XXXVIII.— 7?? inscribe a polygon of any number of sides within a circle, Fig. 52. — Draw the diameter a b, and through the centre E draw the perpendicular e c, cutting the circle at f. Divide e f into four equal parts, and set off three parts equal to those from f to c. Divide the diameter a b into as many equal parts as the polygon is to have sides; and from c draw c d through the second point of division, cutting the circle at d. Then a d is equal to one ON THE ELLIPSE. 13 side of the polygon, and by stepping round the circumference with the Fig. 52.— Proh. XXXVIII. length A D, the polygon may be com- pleted; The constructions for inscribing regular polygons in circles are suit- able also for dividing the circumfer- ence of a circle into a number of equal parts. To supply a means of dividing the circumference into any number of parts, including cases not provided for in the foregoing prob- lems, the annexed table of angles relating to polygons, expressed in dqgrees, will be found of general utib'ty. In tliis table the angle at Table of Polygonal Angles. Number Angle Number Angle , ofSidcs. at Centre. of Sides. at Centre. No. Degrees. No. Degrees. 3 120 12 30 4 90 13 27A 5 72 H 25f 6 60 15 24 7 5if 16 22^ 8 45 17 «^ 9 40 18 20 10 1 36 19 19 II 1 32A 20 18 the centre is found by dividing 360°, the number of degrees in a circle, by the number of sides in the polygon ; and by setting off round the centre of the circle a succession of angles by means of the protractor, equal to the angle in the table due to a given number of sides, the radii so drawn will divide die circimiference into the same number of parts. The triangles thus formed are termed the elemen- tary triangles of the polygon. •Problem XXXIX. — To inscribe any regular polygon in a given circle; or to divide the circumference into a ^ven number of equal parts, by means of ihe angle at the centre. Fig. 53. — Fig. 53.-Prob. XXXIX. Suppose the circle is to contain a hexagon, or is to be divided at the circumference into six equal parts. Find the angle at the centre for a hexagon, or 60°; draw any radius b c, and set off, by a protractor or other- wise, the angle at the centre cbd equal to 60°; then the interval cd is one side of the figure, 6x segment of the circumference; and the remaining points of division maybe found either by stepping along the circumference with the distance c d in the dividers, or by setting off the remaining five angles, of 60^ each, round the centre. PROBLEMS ON THE ELLIPSE. An ellipse is an oval figure, like a circle in perspective. The line a b, Fig. 54, that divides it equally in the direction of its greatest dimension, is the transverse axis; and the per- pendicular CD, through the centre, is the conjugate axis. Two points, F, G, in the transverse axis, are the 14 GEOMETRICAL PROBLEMS foci of the curve, each being called a focus; being so placed that the sum of their distances from either end of the conjugate axis, c or d, is equal F'g- 54* — Prob. XL. to the transverse axis. In general, the sum of their distances from any other point in the curve is equal to the transverse axis. A line drawn at right angles to either axis, and termi- nated by the curve, is a double ordi- nate^ and each half of it is an ordinate. The segments of an axis between an ordinate and its vertices are called abscisses. The double ordinate drawn through a focus is called the para- ineter of the axis. The squares of any two ordinates to the transverse axis, are to each other as the rectangles of their respec- tive abscisses. Problem XL. — To describe an el- lipse when the length and breadth are ^veny Fig. 54. — On the centre c, with A £ as radius, cut the axis a b at f and G, the foci ; fix a couple of pins into the axis at f and G, and loop on a thread or cord upon them equal in length to the axis a b, so as when stretched to reach to the extremity c of the conjugate axis, as shown in dot- lining. Place a pencil or drawpoint inside the cord, as at h, and guiding the pencil in this way, keeping the cord equally in tension, carry the pencil round the pins f, g, and so describe the ellipse. Note, — This method is employed in setting off elliptical garden-plots, walks, &c. 2d Method, Fig. 55. — Along the straight edge of a slip of stiff paper, mark off a distance a c equal to a c, Piff- 55.— Prob. XL. 2d method. I half the. transverse axis; and from the same point a distance a b equal to c D, half the conjugate axis. Place the slip so as to bring the point b on the line a b of the transverse axis, and the points on the line de; and set off on the dra\ving the position of the point a. Shifting the slip, so that the point b travels on the transverse axis, and the point c on the conjugate axis, any number of points in the curve may be found, through which the curve may be traced. id Metlwd^ Fig. 56. — ^The action of the preceding method may be em- Fig. 56.— -Prob. XL. 3d method. bodied so as to afford the means of describing a large curve continuously, by means of a bar mky with steel points w, /, k, rivetted into brass slides adjusted to the length of the semi- axes, and fixed with setscrews. A rec- tangular cross E G, with guiding slots, is placed coinciding with the two ON THE ELLIPSE. IS axes of the ellipse, ac and bh; by sliding the points ky /, in the slots, and carrying round the point m, the curve may be continuously described. A pen or pencil may be fixed at m, 4M Method, Fig. 57. — Bisect the transverse axis at c, and through c Fig. 57. — Prob. XL. 4th method. draw the perpendicular d e, making CD and CE each equal to half the conjugate axis. From d or e, with the radius a c, cut the transverse axis at r, f', for the foci. Divide a c into a number of parts at the points i, 2, 3, &c. With the radius a i, on f and f' as centres, describe arcs ; and with the radius b i, on the same centres, cut these arcs as shown. Repeat the operation for the other divisions of the transverse axis. The series of intersections thus made are points in the curve, through which the curve may be traced. 5/>i Method, Fig. 58.— On the two ^••■•' — •« «»«^ ^•» ^^ *-. , D *^ » _^^^^^^ ^^ % » ^^ X • ly * * » 1 * \: \. id ' * * « • V if * ' * • II /• u 1 ■% f * • ! 1 4i«; ? C_ 1 i ' • 1 *v» ' • •• • 1 1 • » • 1 • 1 * * 1 ' * J 1 * f A • ^k • • * t 1 » \ y f '-^Jv--y' J t • ^rs-* •» ! % 'O*^ / \ : '^^^^IL.J' \AL T • • **——_. T of points, a, b, &c., in the circumfer- ence A F B, draw radii cutting the in- ner circle at d, b\ &c. From a, by &c., draw perpendiculars to ab; and from a'y ^', &c., draw parallels to a b, cutting the respective perpendiculars at «, Oy &c. The intersections are points in the curve, through which the curve may be traced. 6/// Metlwdy Fig. 59. — When the transverse and conjugate diameters Fig. 58.— Prob. XL. 5th method. axes A B, D E as diameters, on centre c, describe circles; from a number Fig. 59.— Prob. XL. 6ih method. are given, a b, c d, draw the tangent E F parallel to a b. Produce c d, and on the centre c, with the radius of half a B, describe a semicircle hdk; from the centre g draw any number of straight lines to the points e, r, &c., in the line e f, cutting the cir- cumference at /, w,«, &c.; from the centre o of the ellipse draw straight lines to the points e, r, &c., and from the points /, w, «, &c., draw parallels to g c, cutting the lines o e, or, &c., at L, M, N, &C. These are points in the circumference of the ellipse, and the curve may be traced through them. Points in the other half of the ellipse are formed by ex- tending the intersecting lines as indi- cated in the figure. Problem XLL — To describe an ellipse approximately by means of cir- cular arcs. — First, with . arcs of two radii. Fig. 60. — Find the difference i6 GEOMETRICAL PROBLEMS of the two axes, and set it off from the centre o to a and r, on oa and oc; Fig^. 60.— IVob. XLI. draw a c, and set off half ac to d; draw di parallel to ac^ set off o^ equal to o//, join ei, and draw the parallels e m, d m. From m, with radius m c, describe an arc through c; and from / describe an arc through D ; from d and e describe arcs through A and B. The four arcs form the ellipse approximately. Note. — ^This method does not ap- ply satisfactorily when the conjugate axis is less than two-thirds of the transverse axis. o M equal to c l, and on d describe an arc with radius dm; on a, with radius o l, cut this arc at a. Thus the five centres d, a, by h, h' are found, from which the arcs are described to form the ellipse. Note, — ^This process works well for nearly all proportions of ellipses. It is employed in striking out vaults and stone bridges. Problem XLII. — To draw a tan- 'K o'. %^» / \ V. Fig. 6x. — Prob. XLI. ad method. Second, with arcs of three radii, Fig. 61. — On the transverse axis ab draw the rectangle b g, on the height o c ; to the diagonal a c draw the per- pendicular ghd; set off ok equal to c, and describe a semicircle on AK, and produce oc to l; set off Fig. 62. -Prob. XLII. gefit to an ellipse through a given point in theatrve. Fig. 62. — From the given point T draw straight lines to the foci F, f'; produce f t beyond the curve to c, and bisect the exterior angle ^ t f, by the line t //, which is the tangent. Problem XLIII. — To draw a tangent to an ellipse from a given point without the curve. Fig. 63. — From the given point t, with a radius to the nearest focus f, de- scribe an arc on the other focus ' f', with a radius equal to the trans- verse axis, cut the arc at k l, and Fig. 63.-Prob. XLIII. draw K f', L f', cutting the curve at M, N. The lines t m, t n are tangents. ON THE PARABOLA. 17 PROBLEMS ON THE PARABOLA. A parabola, dac, Fig. 64, is a cun'e such that every point in the curve is equally distant from the di- rectrix K L and the focus r. The focus lies in the axis a b drawn from the v€rtex or head of the curve a, so as to divide the figure into two equal parts. The vertex a is equidistant from the directrix and the focus, or A^=AF. Any line parallel to the axis is a diameter. A straight line, as EG or D c, drawn across the figure at right angles to the axis is a double ordinate, and either half of it is ap ordinate. The ordinate to the axis E F G, drawn through the focus, is called iki't parameter of the axis. A s^ment of the axis, reckoned from the vertex, is an absciss of the axis; and it is an absciss of the ordinate drawn from the base of the absciss. Thus, A B is an absciss of the ordinate b c. Abscisses of a parabola are as the squares of their ordinates. Problem XLIV. — To describe a parabola when an absciss and its ordi- nate are given; that is to say, when the height and breadth are given, Fig. 64. — Bisect the given ordinate < ^ J, a. J ^ \, /• s \ \ / \ V / 3 $ \ :» J 1 <» c rig. 64.— Prob. XLIV. fic at tf ; draw A a, and then a b per- pendicular to it, meeting the axis at A Set off A if, A F, each equal to b b; and draw k ^ l perpendicular to the aiis. Then k l is the directrix and F is (he focus. Through f and any number of points, Oy Oy &c., in the axis, draw double ordinates, n «, &c. ; and on the centre f, with tlie radii ¥ e,oe, &c., cut the respective ordinates at e, g, «, «, &c. The curve may be traced through these points as shown. 2d Method; by means of a square and a cord. Fig. 65. — Place a straight- Fig. 65.— Prob. XLIV. ad method. edge to the directrix e n, and apply to it a square leg. Fasten to the end G, one end of a thread or cord equal in length to the edge £ g, and attach the other end to the focus f; slide the square along the straight- edge, holding the cord taut against the edge of the square by a draw- point or pencil d, by which the curve is described. ^d Method; when the height and the base are given, Fig. 66. — Let a b be J5 p A F a J r tr ^ V Of ^ y r 1 V I c A f ^ \ ^ 1. m ^ \ a t ^ i f t V 1 ^ C PI r < 1 1 \ 1 ? < i 1 > Fig. 66.— Prob. XLIV. 3d method. the given axis, and c d a double ordi- nate or base; to describe a parabola 2 I8 GEOMETRICAL PROBLEMS of which the vertex passes through a. Through a draw e f parallel to c d, and through c and d draw c e and D F parallel to the axis. Divide b c and BD into any number of equal parts, say five, at «, by &c., and divide c £ and D F into the same number of parts. Through the points a^ b, c, d in the base c d, on each side of the axis, draw perpendiculars, and through a, by Cy //, in c £ and d f, draw lines to the vertex a, cutting the perpendicu- lars at eyfygy h. These are points in the parabola, and the curve cad may be traced as shown, passing through them. PROBLEMS ON THE HYPERBOLA. The vertices a, b. Fig. 67, of oppo- site hyperbolas, are the heads of the curves, and are points in their centre or axial lines. The transverse axis A B is the distance between the ver- tices, of which the centre c is the centre. The conjugate axis g h is a straight line drawn through the centre at right angles to the transverse axis. An ordinate f k is a straight line drawn from any point of the curve perpendicular to the axis. The seg- ments of the transverse axis a f, b f, between an ordinate f k and the ver- tices of the curves, are abscisses. The parameter is the double ordinate drawn through the focus. The as- symptotes are two straight lines, s s, R R, drawn from the centre through the ends of a tangent ed at the vertex, equal and parallel to the conjugate axis, and bisected by the transverse axis. The nature of the hyperbola is such that the difference of the distances of any point in the curve from the foci is always the same, and is equal to the transverse axis. In a hyperbola the squares of any two ordinates to the transverse axes are to each other as the rectangles of their abscisses. Problem XLV. — To describe a hyperbolay the transverse and conjugate axes being givetiy Fig. 67. — Draw ab Fig. 67.— Prob. XLV. equal to the transverse axis, and d e perpendicular to it and equal to the conjugate G h. On c, with the radius c E, describe a circle cutting a b pro- duced, at f/; these points are the foci. In A B produced take any number of points Oy Oy &c., with the radii a^^ b^, and on centres f,/ describe arcs cut- ting each other at «, ;z, &c. These are points in the curve, through which it may be traced. 2d Metliody Fig. 67. — The curve may be drawn thus: — Let the ends of two threads/p q, f p q, be fastened at the points /, f, and be made to pass through a small bead or pin p, and knotted together at q. Take hold of Q, and draw the threads tight ; move the bead along the threads, and the point ? will describe the curve. If the end of the long thread be fixed at F, and the short thread at /, the opposite curve may be described in the same ftianner. Or, the line /q may be replaced by a straight-edge turning on a pin at/ and the cord F q joined to it at Q. The curve may then be described by means of a point or pencil in the same manner as for the parabola. Fig. 65. 3^/ Method; when the breadth c d. ON THE HYPERBOLA, CYCLOID, EPICYCLOID. 19 hd^ A B, and transverse axis a a! of the curve are gtvetiy Fig. 68. — Divide Fig. 68.— Protx XLV. 3d method. the base or double ordinate c d into a number of equal parts on each side of the axis at a, d, &c ; and divide the parallels c e, d f, into the same number of equal parts at a, d, &c. From the points a, d, &c., in the base, draw lines to a', and from the points a, b, &c., in the verticals, draw lines to A, cutting the respective lines from the base. Trace the curve through the intersections thus obtained. THE CYCLOID AND EPICYCLOID. Problem XLVL — To describe a cydoid^ Fig. 69. — When a wheel or a circle D G c rolls along a straight line Fig. 69.— Prob. XLVI. one revolution, it measures off a straight line a b exactly equal to the circumference of the circle d g c, which is called the generating circle, and a point or pencil fixed at the point D in the circumference traces out a curvilinear path a d b, called a cycloid, A B is the bcLse and c D is the axis of the cycloid. Place the generating circle in the middle of the cycloid, as in the figure, draw a line e h parallel to the base, cutting the circle at g; and the tan- gent H I to the curve at the point h. Then the following are some of the properties of the cycloid : — The horizontal line h G=arc of the circle G d. The half-base a c =» the half-circum- ference c G D. The arc of the cycloid d h = twice the chord d g. The half-arc of the cycloid d a = twice the diameter of the circle d c. Or, the whole arc of the cycloid A D B = four times the axis c d. The area of the cycloid a d b a = three times the area of the generating circle d c The tangent H i is parallel to the chord G D. Problem XLVII. — To describe an Fig. 7o.-Prob. XLVIL A B, Fig. 69, beginning at a and end- exterior epicycloid ^ Fig. 70. — The epicy- ing at D, where it has just completed chid differs from the cycloid in this, 20 GEOMETRICAL PROBLEMS that it is generated by a point d in one circle do rolling upon the cir- cumference of another circle a c b, instead of on a flat surface or line; the former being the generating circle^ and the latter the fundametital circle. The generating circle is shown in four positions, in which the generating point is successively marked d, d', d", d'^'. a d"' b is the epicycloid. Problem XLVIIL — To describe Fig. 71.— Prob. XLVIIL an interior epicycloid^ Fig. 71. — If the generating circle be rolled on the in- side of the fundamental circle, as in Fig. 71, it forms an interior epicycloid^ or hypocycloidy a j>"' b, which becomes in this case nearly a straight line. The other points of reference in the figure correspond to those in Fig. 70. When the diameter of the generating circle is equal to half that of the fun- damental circle, the epicycloid be- comes a straight line, being in fact a diameter of the larger circle. THE CATENARY. « When a perfecdy flexible string, or a chain consisting of short links, is suspended from two points m, n, Fig. 72, it is stretched by its own weight, and it forms a curve line known as the catenary, m c n. The point c, where the catenary is horizontal, is the vertex. Problem XLIX. — To describe a catenary^ Fig. 72. — Draw the vertical c G equal to the length of the arc of the chain, m c, on one side of the vertex, and divide it into a great number of equal parts,at ( i ), ( 2), (3 ),&c. Draw the horizontal line c h equal to the length of so much of the rope or chain as measures by its weight the horizontal tension of the chain. From the point c as the vertex, set off" c (i) on the horizontal line equal to c i on the vertical; and (i) (2) from the point (i), parallel to h i and equal to c(i); and again (2) (3) from the point (2) parallel to h 2 and equal to c (i); and so on till the last segment (6) m is drawn parallel to h g. The poly- gon c (i) (2) (3) . . . M, thus formed, is approximately the catenary curve, which may be traced through the middle points of the sides of the polygon. A similar process being performed for the other side of the curve, the catenary is completed. Fig. 7a.— Prob. XLIX. 2d Method. — Suspend a finely linked chain against a vertical wall. The curve may be traced from it, on the wall, ^swering the conditions of given length and height, or of given width or length of arc, A cord having numerous equal weights suspended from it at short and equal distances may be used. CIRCLES, PLANE TRIGONOMETRY. 21 CIRCLES. The circumference of a circle is commonly signified in mathematical discussions by the symbol x, which indicates the length of the circumfer- ence when the diameter t= i. The area of a circle is as the square of the diameter, or the square of the circumference. The ratio of the diameter to the circumference is as i to 3*141593 — commonly abbreviated, as i to 3'i4i6 approximately, as i to 3I or as 7 to 22 WTien the diameter = i, the area is equal to 785398 + or, commonly abbreviated, 7854 approximately, j^ths. HTien the circumference = i, the area is equal to "079577 + or, abbreviated, , '0796 approximately, A^hs, or '08. In these ratios, the diameter and the circumference are taken lineally, and the area superficially. So that if the diameter = i foot, the circum- ference is equal to 3'i4i6 feet, and the area is equal to 7854 square foot Note, — If the first three odd figures, 1,3, 5, be each put down twice, the first three of these will be to the last three, that is 113 is to 355, as the diameter to the circumference. PLANE TRIGONOMETRY. The circumference of a circle is supposed to be divided into 360 degrees or divisions, and as the total angularity about the centre is equal to four right angles, each right angle contains 90 degrees, or 90°, and half a right angle <- Fig. 73. — Definidons in Plane Trigonometry. contains 45**. Each degree is divided into 60 minutes, or 6o'j and, for the sake of still further minuteness of measurement, each minute is divided into 60 seconds, or 60", In a whole circle there are, therefore, 360 x 60 x 60 = 22 GEOMETRICAL PROBLEMS. 1,296,000 seconds. The annexed diagram, Fig. 73, exemplifies the rela- tive positions of the sine, cosine, versed sine, tangent, co-tangent, secant, and co-secant of an angle. It may be stated, generally, that the correlated quantities, namely, the cosine, co-tangent, and co-secant of an angle, are the sine, tangent, and secant, respectively, of the complement of the given angle, the complement being the difference between the given angle and a right angle. The supplement of an angle is the amount by which it is less than two right angles. When the sines and cosines of angles have been calculated (by means of formulas which it is not necessary here to particularize), the tangents, co-tan- gents, secants, and co-secants are deduced from them according to the following relations : — rad. X sin. rad/^ rad.*^ rad.^ tan. = ; cotan. = ; sec. = ; cosec. = . COS. tan. COS. sin. For these the values will be amplified in tabular form. A triangle consists of three sides and three angles. When any three of these are given, including a side, the other three may be found by cal- culation : — , Case i. — IVAm a side and its opposite angle are two of the ^vm parts. Rule i. To find a side, work the following proportion: — as the sine of the angle opposite the given side is to the sine of the angle opposite the required side, so is the given side to the required side. Rule 2. To find an angle: — as the side opposite to the given angle is to the side opposite to the required angle, so is the sine of the given angle to the sine of the required angle. Rule 3. In a right-angled triangle, when the angles and ofie side tiext t/ie right angle are given, to find the other side: — as radius is to the tangent of the angle adjacent to the given side, so is this side to the other side. Case 2. — When two sides and the included angle are given. Rule 4. To find the other side: — as the sum of the two given sides is to their difference, so is the tangent of half the sum of their opposite angles to the tangent of half their difference — add this half difference to the half sum, to find the greater angle; and subtract the half difference from the half sum, to find the less angle. Tlie other side may then be found by Rule i. Rule 5. When the sides of a right-angled triangle are givai, to find the angles: — , MENSURATION OF SURFACES. 23 as one side is to the other side, so is the radius to the tangent of the angle adjacent to the first side. Case 3. — When the three sides are given. Rule 6. To find an angle. Subtract the sum of the logarithms of the sides which contain the required angle, from 20; to the remainder add the logarithm of half the sum of the three sides, and that of the difference bet^s'ccn this half sum and the side opposite to the required angle. Half the sum of these three logarithms will be the logarithmic cosine of half the required angle. The other angles may be found by Rule i. Rule 7. Subtract the sum of the logarithms of the two sides which con- tain the required angle, from 20, and to the remainder add the logarithms of the differences between these two sides and half the sum of the three sides. Half the result will be the logarithmic sine of half the required angle. Note, — In all ordinary cases either of these rules gives sufficiently accur- ate results. It is recommended that Rule 6 should be used when the required angle exceeds 90°; and Rule 7 when it is less than 90°. MENSURATION OF SURFACES. To find the area of a paraiieiogram. Multiply the length by the height, or perpendicular breadth. Or, multiply the product of two contiguous sides by the natural sine of the included angle. To find the area of a triangle. Multiply the base by the perpendicular height, and take half the product Or, multiply half the product of two contiguous sides by the natural sine of the included angle. To find the area of a trapezoid. Multiply half the sum of the parallel sides by the perpendicular distance between them. To find the area of a quadrilateral inscribed in a circle. From half the sum of the four sides subtract each side severally; multiply the four re- mainders together; the square root of the product is the area. To find the area of any qiiadrilateral figure. Divide the quadrilateral into two triangles; the sum of the areas of the triangles is the area. Or, midriply half the product of the two diagonals by the natural sine of the angle at their intersection. Note, — As the diagonals of a square and a rhombus intersect at right angles (the natural sine of which is i), half the product of their diagonals is the area. To find the area of any polygon. Divide the polygon into triangles and trapezoids by drawing diagonals; find the areas of these as above shown, for the area. To find the area of a regular polygon. Multiply half the perimeter of the polygon by the perpendicular drawn from the centre to one of the sides. Nc^e, — ^To find the perpendicular when the side is given — 24 GEOMETRICAL PROBLEMS. as radius to tangent of half-angle at perimeter (see table No. i), SO is half length of side to perpendicular. Or, multiply the square of a side of any regular polygon by the corres- ponding area^in the following table: — Table No. i. — ^Angles and Areas of Regular Polygons, Name. Number of Sides. One half Angle at the Perimeter. Area. (Side=i) 1 Perpendi- cular. (Side = I) Equilateral triangle, Square, 3 4 5 6 7 8 9 lO II 12 30° 45° 54° 60° 64°l 67°i 70° 72° 73°A 75° 0-4330 I -oooo 17205 2-5981 3*6339 4-8284 6I8I8 76942 93656 II-I962 0-2887 0-5000 0-6882 0-8660 1*0383 1-2071 1*3737 1-5388 17028 I -8660 .uri|^i.M>&^, .•■.«•. Pentagon Hexagon Heptagon Octagon Nonagon Decagon Undecagon Dodecagon To find the circumference of a circle. Multiply the diameter by 3 -14 16. Or, multiply the area by 12*5664; the square root of the product is the circumference. To find the diameter of a circle. Divide the circumference by 3-1416. Or, multiply the circumference by '3183. Or, divide the area by 7854; the square root of the quotient is the diameter. To find the area of a circle. Multiply the square of the diameter by 7854. Or, multiply the circumference by one-fourth of the diameter. Or, multiply the square of the circumference by '07958. To find the length of an arc of a circle. Multiply the number of degrees in the arc by the radius, and by '01745. Or, the length may be found nearly, by subtracting the chord of the whole arc from eight times the chord of half the arc, and taking one-third of the remainder. To find the area of a sector of a circle. Multiply half the length of the arc of the sector by the radius. Or, multiply the number of degrees in the arc by the square of the radius, and by -008727. To find the area of a segment of a circle. Find the area of the sector which has the same arc as the segment; also the area of the triangle formed by the radial sides of the sector and the chord of the arc; the difference or the sum of these areas will be the area of the segment, ac- cording as it is less or greater than a semicircle. To find the area of a ring included between the circumferences of two con- MENSURATION OF SURFACES. 25 oniric circles. Multiply the sum of the diameters by their difference, and 577854. • To find Ike area of a cycloid. Multiply the area of the generating circle To find tJu length of an arc of a parabola^ cut off by a double ordinate to the axis. To the square of the ordinate add four-fifths of the square of the absciss; twice the square root of the sum is the length nearly. . Note. — ^This rule is an approximation which applies to those cases only in which the absciss does not exceed half the ordinate. To find the area of a parabola. Multiply the base by the height; two- thirds of the product is the area. To find the circumference of an ellipse. Multiply the square root of half the sum of the squares of the two axes by 3*1416. To find the area of an ellipse. Multiply the product of the two axes by 7^54. Note, — ^The area of an ellipse is equal to the area of a circle of which the diameter is a mean proportional between the two axes. To find the area of an elliptic segment, the base of which is parallel to either axis of the ellipse. Divide the height of the segment by the axis of which it is a part, and find the area of a circular segment, by table No. VII., of which the height is equal to this quotient; multiply the area thus found by the two axes of the ellipse successively; the product is the area. To find the length of cm arc of a hyperbola^ beginning at the vertex. To 19 times the transverse axis add 21 times the parameter to this axis, and multiply the sum by the quotient of the absciss divided by the transverse. 2(L To 9 times the transverse add 2 1 times the parameter, and multiply the sum by the quotient of the absciss divided by the transverse. 3d. To each of these products add 15 times the parameter, and then as the latter sum is to the former sum, so is the ordinate to the length of the arc, nearly. To find the area of a hyperbola. To the product of the transverse and absciss add five-sevenths of the square of the absciss, and multiply the square root of the sum by 21; to this product add 4 times the square root of the product of the transverse and absciss; multiply the sum by 4 times the product of the conjugate and absciss, and divide by 75 times the transverse. The quotient is the area nearly. To find the area of any cundlinecU figure, hounded at the ends by parallel straight lines, Fig. 74. Divide the length of the figure ab into any even number of equal parts, and draw oidinates c, d, e, &c, through the pomts of division, to touch the boundary lines. Add together the first and last ordinates {c and k), and call the sum a; add together the even ordinates (that is, ^Jy ^y ), and call the sum b; add together the odd ordinates, except the first and last (e,g, i), and call the sum c Let D be the common distance of the ordinates, then Fig. 74.— For Area of Curvilinear Figure. 26 GEOMETRICAL PROBLEMS. (a + 4 B + 2 c) X D = area of figure. This IS known as Simpson's Rule. 2d Method^ Fig. 74. — Having divided the figure into an even or an odd number of equal parts, add together the first and last ordinates, making the sum a; and add together all the intermediate ordinates, making the sum B. Let l = the length of the figure, and n = the number of divisions, then A + 2B 2n X L = area of figure. That is to say, twice the sum of the intermediate ordinates, plus the first and last ordinates, divided by twice the number of divisions, and multi- plied by the length, is equal to the area of the figure. This method is that commonly used; it is sufficiently near to exactness for most purposes. 3^ Methody Fig. 74. — Having divided the' figure as above, measure by a scale the mean depth of each division, at the middle of the division; add together the depths of all the divisions, and divide the sum by the number of divisions, for the average depth; multiply the average depth by the length, which gives the area. For the sake of obtaining a more nearly exact result, the figure may be divided into two half-parts, c^k^ Fig. 75, one at each end, and a number of whole equal parts, d^e^f^gji^ij^ intermediately. Then the ordinates separating these parts, excluding the extreme ordinates, may be measured » Fig. 75. For- Area of Cunrilincal Figures. Fig. 76. direct, and the sum of the measurements divided by the nimiber of them, and multiplied by the length, for the area. Note, — In dealing with figures of excessively irregular outline, as in Fig. 76, representing an indicator-diagram from a steam-engine, mean Hnes, ab^ c dy may be substituted for the actual lines, being so traced as to intersect the undulations, so that the total area of the spaces cut off may be com- pensated by that of the extra spaces inclosed. Note 2. — The figures have been supposed to be bounded at the ends by parallel planes. But they may be terminated by curves or angles, as in Fig. 76, at ^, when the extreme ordinates become nothing. MENSURATION OF SOLIDS. 2^ MENSURATION OF SOLIDS. To find tke surface of a prism or a cylinder. The perimeter of the end multiplied by the height gives the upright surface ; add twice the area of an end. Tofijid the cubic contents of a prism or a cylinder. Multiply the area of the base by the height To find the surface of a pyramid or a cone. Multiply the perimeter of the base by half the slant height, and add the area of the base. To find the aibic contents of a pyramid or a cone. Multiply the area of the base by one-third of the perpendicular height. To find the surface of afrustrum of a pyramid or a cone. Multiply the sum of the perimeters of the ends by half the slant height, and add the areas of the ends. To find the cubic cofitents of a frustrum of a pyramid, or a cone, — Add together the areas of the two ends, and the mean proportional between them (that is, the square root of their product), and multiply the sum by one-third of the perp>endicular height. Or, when the ends are circles, add together the square of each diameter, and the product of the diameters, and multiply the sum by 7854, and by one-third of the height To find the aibic contaits of a wedge, — To twice the length of the base add the length of the edge ; multiply the sum by the breadth of the base, and by one-sixth of the height To find the cubic contents of aprismoid {a solid of which the tivo etuis are dis- similar but parallel plane figures of the same number of sides), — To the sum of the areas of the two ends, add four times the area of a section parallel to and equally distant from both ends; and multiply the sum by one-sixth of the length Note. — ^This rule gives the true content of all fmstrums, and of all solids of which the parallel sections are similar figures; and is a good approxima- tion for other kinds of areas and solidities. To find the surface of a sphere. — Multiply the square of the diameter by 31416. Note. — ^The surface of a sphere is equal to 4 times the area of one of its great circles. 2. The surface of a sphere is equal to the convex surface of its circum- scribing cylinder. 3. The surfaces of spheres are to one another as the squares of their diameters. To find the curve surface of any segmmt or zone of a sphere. — Multiply the diameter of the sphere by the height of the zone or segment, and by 3*1416. Note. — The curve surfaces of segments or zones of the same sphere are to one another as their heights. To find the cubic contents of a sphere, — Multiply the cube of the diameter by -5236. Or, multiply the surface by one-sixth of the diameter. 28 GEOMETRICAL PROBLEMS. Note. — The contents of a sphere are two-thirds of the contents of its circumscribing cylinder. 2. The contents of spheres are to one another as the cubes of their diameters. To find the aibic contents of a segment of a sphere. — From 3 times the diameter of the sphere subtract twice the height of the segment; multiply the difference by the square of the height, and by '5236. Or, to 3 times the square of the radius of the base of the segment, add the square of its height; and multiply the sum by the height, and by '5236. To find the cubic contents of a frustrum or zone of a sphere, — ^To the sum of the squares of the radii of the ends add Yi of the square of the height; multiply the sum by the height, and by 1*5708. To find the cubic contents of a spheroid. — Multiply the square of the re- volving axis by the fixed axis and by '5236. Note. — ^The contents of a spheroid are two-thirds of the contents of its circumscribing cylinder. 2. If the fixed and revolving axes of an oblate spheroid be equal to the revolving and fixed axes of an oblong spheroid respectively, the contents of the oblate are to those of the oblong spheroid as the greater to the less axis. To find the cubic contents of a segment of a spheroid. — ist. When the base is parallel to the revolving axis. Multiply the difference between thrice the fixed axis and double the height of the segment, by the square of the height, and the product by '5236. Then, as the square of the fixed axis is to the square of the revolving axis, so is the last product to the content of the segment 2d. When the base is perpendicular to the revolving axis. Multiply the diflference between thrice the revolving axis and double the height of the segment, by the square of the height, and the product by '5236. Then, as the revolving axis is to the fixed axis, so is the last product to the content of the segment. To find tlie solidity of the middle frustrum of a spheroid. — ist When the ends are circular, or parallel to the revolving axis. To twice the square of the middle diameter, add the square of the diameter of one end; multiply the sum by the length of the frustrum, and the product by '2618 for the content. 2d. When the ends are elliptical, or perpendicular to the revolving axis. To twice the product of the transverse and conjugate diameters of the middle section, add the product of the transverse and conjugate diameters of one end; multiply the sum by the length of the frustrum, and by '2618 for the content. To find tJie cubic contents of a parabolic conoid. — Multiply the area of the base by half the height. Or, multiply the square of the diameter of the base by the height, and by -3927. To find the cubic contents of a frustrum of a parabolic cofioid. — Multiply half the sum of the areas of the two ends by the height of the frustrum. MENSURATION OF SOLIDS. 29 Or, muUiply the sum of the squares of the diameters of the two ends by the height, and by '3927. Tofifid the cubic contents of a parabolic spindle, — Multiply the square of the middle diameter by the length, and by -41888. To find the cubic contents of the middle frustrum of a parabolic spindle, — Add together 8 times the square of the largest diameter, 3 times the square of the diameter at the ends, and 4 times the product of the diameters; multiply the sum by the length of the frustrum, and by '05236. To find the surface and the cubic contmts of any of the five regular solids^ Figs. Fig- 77- Fig. 78. Fig. 79- Fig. 80. Fig. 81. 77, 78, 79, 80, 81. — For the surface, multiply the tabular area below, by the square of the edge of the solid. For the contents, multiply the tabular contents below, by the cube of the given edge. Note. — ^A regular solid is bounded by similar and regular plane figures. There are five regular solids, shown by Figs. 77 to 81, namely: — The tetrahedron^ bounded by four equilateral triangles. The hexahedron, or cube, bounded by six squares. The octahedron^ bounded by eight equilateral triangles. The dadecahedron,\>o\ixidtd by twelve pentagons. The icosahedron^ bounded by twenty equilateral triangles. Regular solids may be circumscribed by spheres; and spheres may be inscribed in regular solids. Surfaces and Cubic Contents of Regular Solids. Number of sides. Name. Area. Edge = I. Contents. Edge=i. 4 6 8 12 20 Tetrahedron Hexahedron Octahedron '. . Dodecahedron Icosahedron 1:7320 6*0000 3*4641 20*6458 86603 0*1178 I *oooo 0*4714 7*6631 2*1817 To find the cubic contents of an irregular solid, — Suppose it divided into parts, resembling prisms or other bodies measurable by preceding rules; find the content of each part; the sum of the contents is the cubic contents of the solid. Note, — The content of a small part is found nearly by multiplying half the sum of the areas of each end by the perpendicular distance between them. 30 GEOMETRICAL PROBLEMS. Or, the contents of small irregular solids may sometimes be found by im- mersing them under water in a prismatic or cylindrical vessel, and observing the amount by which the level of the water descends when the solid is withdrawn. The sectional area of the vessel being multiplied by the descent of the level, gives the cubic contents. Or, when the solid is very large, and a great degree of accuracy is not requisite, measure its length, breadth, and depth in several different places, and take the mean of the measurement for each dimension, and multiply the three means together. Or, when the surface of the solid is very extensive, it is better to divide it into triangles, to find the area of each triangle,' and to multiply it by the mean depth of the triangle for the contents of each triangular portion ; the contents of the triangular sections are to be added together. The mean depth of a triangular section is obtained by measuring the depth at each angle, adding . together the three measurements, and taking one-third of the sum. MENSURATION OF HEIGHTS AND DISTANCES. To find the height of an accessible object. — Measure the distance from the base of the object to any convenient station on the same horizontal plane; and at this station take the angle of altitude. Then as radius to tangent of the angle of altitude, so is the horizontal distance to the height of the object above the horizontal plane passing through the eye of the observer. Add the height of the eye, and the sum is the height of the object. Note, — The station should be chosen so that the angle of altitude should be as near to 45° as practicable; because the nearer to 45'', the less is the error in altitude arising from error of observation. When the angle of elevation is 45^ the height above the plane of the eye is equal to the distance. When it is 26° 34', the height is half the dis- tance. To find approximately the height of an accessible object. — There are four methods based on the principle of similar triangles. I St. By a geometrical square^ Fig. 82. — This is a square, a b, with two sights on one of its sides, a ;/, a plumb-line hung from one extremity, ;/, of that side, and each of the twp sides opposite to that extremity, mb,ma^ divided into 100 equal parts; the division beginning at the remote ends, so that the 1 00th divisions meet at the corner m. Let re be the object, and the sights be directed to the summit ^, at the known distance ad. When the Fig. 82.-Mcnsuration of a plummet cuts the side b m at, say, c, then by similar triangles, nb\nc\\ad\de. Or, if the plumb-line cuts the side a m, then the part of a m cut off is to <z « : : ad\ de. Adding to de the height of the eye rd, the sum is the height of the object, re. MENSURATION OF HEIGHTS AND DISTANCES. 31 2d. By shadows. Fig. 83.~7When the sun shines, fix a pole ^^ in the ground, vertically, and measure its shadow a b. Measure also the shadow de Fig. 83. Mensuration of a Height. of the object € m; then, by similar triangles, ab',bc\:de\ e m^ the height of the object. 3d. By r^4ctiofiy Fig. 84. — Place a basin of water, or any horizontal reflecting surface, at a, level with the base of the object de, and retire from it till the eye at c sees the top of the object e, in the centre of the basin at a. Then, by similar triangles, abi bc\\ad\de, 4tt By two poles. Fig. 85. — Fix two poles a m, cfiy of unequal lengths, parallel to the object er, so that the eye of the observer at a, the top of the shorter f>ole, may see c, the top of the longer pole, in a line with e, the summit of the object re. By similar triangles, ab \bc\\ad\de\ and adding rd, the height of the eye, to de, the sum r^ is the height of the object. To find the distance of the visible horizon. — To half the logarithm of the height of the eye, add 3*8105; the sum is the logarithm of the distance in feet, nearly. To find the distance of an object by the motion of sound, — Multiply the number of seconds that elapse between the flash or other sign of the gene- ration of the sound and the arrival of the sound to the ear, by 1120. The product is the distance in feet. Note. — ^\Vhen a sound generated near the ear returns as an echo, half the interval of time is to be taken, to find the distance of the reflecting surface. Fig. 85. Menmration of a Height. 32 MATHEMATICAL TABLES. MATHEMATICAL TABLES. Table No. I. — Of Logarithms of Numbers from i to 10,000. Logarithms consist of integers and decimals; but, for the sake of com- pactness, the integers have been omitted in the table, except in the short preliminary section containing the complete logarithms of numbers from i to 100. The table No. I. contains the decimal parts, to six places, of the loga- rithms of numbers from i to 10,000. The integer, or index, or character- istic of a logarithm, standing on the left-hand side of the decimal point, is a number less by i than the number of figures or places in the integer of the number. If a number contains both integers and decimals, the index is regulated according to the integers. If it contain only decimals, the index is equal to the number of cyphers next the decimal point, plus i; moreover, the index is negative, and is so distinguished by the sign minus, — , written over it For example, to illustrate the adjustment of the integer of the logarithm to the composition of the number : — Numbar. Logarithm. 4743 3676053 474.3 2.676053 47.43 L676053 4.743 0.676053 .4743 .L676053 .04743 .£.676053 .004743 3676053 Still more for the sake of compactness, the first two figures of the loga- rithms are given only at the beginning of each line of logarithms, to save repetition, only the remaining four decimal places being given for each logarithm. In seeking for a logarithm, the eye readily takes in the prefixed two digits at the commencement of each line. Rules, — To find the logarithm of a number containing one or two digits, look for the number in the preliminary tablet in one of the columns marked No., and find the logarithm next it Or, look in the body of the table for the given number in the columns marked N, with one or two cyphers following it; the decimal part of the logarithm is in the column next to it For example, the decimal part of the logarithm of 3 is found, in the column next to the number 300, to be .477121, and as there is but one digit, the logarithm is completed with a cypher, thus, 0.47 7 12 1. The same logarithm stands for 30, except that, when completed, it becomes 1.477 121. Again, take the number 37; look for 370 in column N, and the decimal part of the logarithm is found, in the colunm next it, to be .568202, which, being completed, becomes 1.568202. If the number be .37, the logarithm becomes 1.568202. To find the logarithm of a number consisting of three digits, look for the EXPLANATION AND USES OF THE TABLES. 33 number in column N, and find the logarithm in the column next it, as already exemplified, for which the index is to be setded and prefixed as before. If the number consist of four digits, look for the first three in column N, and the fourth in the horizontal line at the head or at the foot of the table. The decimal part of the logarithm is found opposite the three first digits and under or over the fourth. Take the number 5432; opposite 543 in column N, and in the column headed 2, is the logarithm .734960, to which 3 is to be prefixed, making 3.734960. If the number be 5.432, the complete logarithm is 0.734960. If the number consist of five or more digits, find the logarithm for the first four as above; multiply the difference, in column D, by the remaining digits, and divide by 10 if there be only one digit more, by 100 if there be two more, and so on; add the quotient to the logarithm for the first four. The sum is the decimal part of the required logarithm, to which the index is to be prefixed. For example, take 3. 141 6. The logarithm of 3 141 is .497068, decimal part; and the difference, 138 x 6 -h 10 = 83, is to be added, thus — 0.497068 83 making the complete logarithm, 0.497151 To find the number corresponding to a given logarithm, look for the logarithm without the index. If it be found exactly or within two or three units of the right-hand digit, then the first three figures of the indicated number will be found in the number column, in a line with the logarithm, and the fourth figure at the top or the foot of the column containing the logarithm. Annex the fourth figure to the first three, and place the decimal ix)int in its proper position, on the principles already explained. If the given logarithm differs by more than two or three units from the nearest in the table, find the number for the next less tabulated logarithm, which will give the four first digits of the required number. To find the fifth and sixth digits, subtract the tabulated logarithm from the given loga- rithm, add two c)rphers, and divide by the difference found in column D opposite the logarithm. Annex the quotient to the four digits already found,. and place the decimal point For example, to find the number represented by the logarithm 2.564732: — 2.564732 given logaridim. Log. 367.0= 2.564666 nearest less. 367.056 56 D 118)6600 (56 nearly. 590 700 708 Showing that the required number is 367.056. To multiply together two or more numbers, add together the logarithms . 3 34 MATHEMATICAL TABLES. of the numbers, and the sum is the logarithm of the product Thus, to multiply 365 by 3.146: — I-og 365 = 2.562293 Log 3.146 = 0.497759 3.060052 Log 1148 3-059942 29 D 380)11000 (29 nearly. 760 1148.29 3400 3420 Showing that the product is 1148.29. To divide one number by another, subtract the logarithm of the divisor from that of the dividend, and the remainder is the logarithm of the quotient To find any power of a given number, multiply the logarithm of the num- ber by the exponent of the power. The product is the logarithm of the power. To find any root of a given number, divide the logarithm of the number by the index of the root The quotient is the logarithm of the root To find the reciprocal of a number, subtract the decimal part of the logarithm of the number from 0.000000; add i to the index of the loga- rithm, and change the sign of the index. This completes the logarithm of the reciprocal. For example, to find the reciprocal of 230: — 0.000000 Log 230 = 2.361728 3.638272= log 0.004348 (reciprocal). Inversely, to find the reciprocal of the decimal .004348 :■— 0.000000 Log .004348 = 3.638272 2.361728 = log 230 (reciprocal). Note. — It will be found in practice, for the most part, unnecessary to note the indices of logarithms, as the decimal parts are in most cases suffi- ciently indicative of the numbers without the indices. The exact calcula- tion of differences may also in most cases be dispensed with — rough mental approximations being sufficiently near for the purpose — particularly when the numbers contain decimals. The indices are, however, indispensable in the calculation of the roots of numbers. EXPLANATION AND USES OF THE TABLES. 35 Tabl^ No. IL — Of Hyperbolic Logarithms of Numbers. In this table, the numbers range from i.oi to 30, advancing by .01, up to the whole number 10; and thence by larger intervals up to 30. The h)'perbolic logarithms of numbers, or Neperian logarithms, as they are sometimes called, are calculated by multiplying the common logarithms of the given numbers, in table No.. I., by the constant multiplier, 2.302585. The hyperbolic logarithms of numbers intermediate between those which are given in the table, may be readily obtained by interpolating proportional differences. Table No. III. — Of Circumferences, Circular Areas, Squares and Cubes; and of Square Roots and Cube Roots. It ha«g been shown how to calculate the powers and roots of numbers by means of logarithms. The table No. III. will be useful for reference. It contains the powers and roots of numbers consecutively from i to 1000. The circumferences and areas of circles, due to the numbers contained in the first columns, considered as diameters, are also given. They will be found useful when diameters are expressed in integers and decimals, or otherwise than in feet, inches, and fractions. Table No. IV. — Of Circumferences and Areas of Circles, with Sides of Equal Squares. The Table No. IV. gives the circumferences and areas of circles from ^ inch to 120 inches in diameter, advancing by sixteenths of an inch up to 6 inches diameter; thence by eighths of an inch to 50 inches diameter; thence by quarters of an inch to 100 inches diameter; and thence by half inches to 120 inches diameter. At the same time, the decimal equivalents of fractions of inches are given in the first columns, and they are complemented by inches and decimals advancing by tenths, for which also the circumferences and areas are given. The table is thus completed for diameters expressed with decimals, as well as for those expressed with vulgar fractions. By a suitable adjustment of decimal points the circumferences and areas may be determined from the contents of the table for diameters ten or a hundred times as much as, or less than, the values given in the first column. ^\Tiilst the diameters are here expressed as inches, they may be taken as feet, or as measures of any other denomination. The column of sides of equal squares^ contains the sides of squares having the same area as the circles in the same hnes of the table respectively. Note, — ^The column oi circular areas given in table No. III., contains the areas of circles of which the diameters are given in common numbers in the first column. Tables Nos. V. and VI. — Of Lengths of Circular Arcs. The lengths of circular arcs are given proportionally to that of the radius, and to that of the chord, in the tables Nos. V. and VL In the first of these tables, the radius is taken = i, and the number of degrees in the arc are given in the first column. The length of the arc as compared with the radius is given decimally in the second column. 36 MATHEMATICAL TABLES. In the second table, the chord is taken = i, and the rise or height of the arc, expressed decimally as compared with the chord, is given in the first column. The length of the arc relatively to the chord is given in the second column. To use the first table, No. V., find the proportional length of the arc corresponding to the degrees in the arc, and multiply it by the actual length of the radius; the product is the actual length of the arc. To use the second table, No. VI., divide the height of the arc by the chord for the proportional height of the arc, which find in the first column of the table ; die proportional length of the arc corresponding to it being multi- plied by the actual length of the chord, gives the actual length of the arc. Note, — The length of an arc of a circle may be found nearly thus: — Subtract the chord of the whole arc from 8 times the chord of half the arc. A third of the remainder is the length nearly. Table No. VII. — Of Areas of Circular Segments. The areas of circular segments are given in Table No. VII., in proportional superficial measure, the diameter of the circle of which the segment forms a portion being = i. The height of the segment, expressed decimally in proportion to the diameter, is given in the first column, and the relative area in the second column. To use the table, divide the height by the diameter, find the quotient in the table, and multiply the corresponding area by the square of the actual length of the diameter; the product will be the actual area. Table No. VIII. — Sines, Cosines, Tangents, Cotangents, Secants, AND Cosecants of Angles from o° to 90° This table, Na VIII., is constructed for angles of from 0° to 90°, advancing by 10', or one-sixth of a degree. The length of the radius is equal to i, and forms the basis for the relative lengths given in the table, and which are given to six places of decimals. Each entry in the table has a duplicate significance, being the sine, tangent, or secant of one angle, and at the same time the cosine, cotangent, or cosecant of its complement For this reason, and for the sake of compactness, the headings of the columns are reversed at the foot; so that the upper headings are correct for the angles named in the left hand mai^n of the table, and the lower headings for those named in the right hand margin. To find the sine, or other element, to odd minutes, divide the difference between the sines, &c., of the two angles greater and less than the given angle, in the same proportion that the given angle divides the difference of the two angles, and add one of the parts to the sine next it By an inverse process the angle may be found for an^ given sine, &c., not found in the table. Table No. IX. — Of Logarithmic Sines, Cosines, Tangents, and Co- tangents of Angles from 0° to 90^ This table. No. IX., is constructed similarly to the table of natural sines, &c., preceding. To avoid the use of logarithms with negative indices, the radius is assumed, instead of being equal to i, to be equal to 10", or EXPLANATION AND USES OF THE TABLES. 37 io,cx)o,ooo,ooo; consequently the logarithm of the radius = lo log lo = lo, ^^Tience, if, to log sine of any angle, when calculated for a radius = i, there be added lo, the sum will be the log sine of that angle for a radius = lo". For example, to find the logarithmic sine of the angle 15° 50'. Nat sine 15'' 50'= -272840; its log = 1*435908 add = 10 Logarithmic sine of 15** 50'= 9*435908 When the logarithmic sines and cosines have been found in this manner, the logarithmic tangents, cotangents, secants, and cosecants are found from those by addition or subtraction, according to the correlations of the trigdnometrical elements already given, and here repeated in logarithmic fonn: — Log tan = 10 + log sin. - log. cosin. Log cotan = 20 — log tan. Log sec. = 20 — log cosin. Log cosec = 20 - log sin. To find the logarithmic sine^ tangent^ &*c., of any angle, — ^When the number of degrees is less than 45°, find the degrees and minutes in the left hand colunm headed angle^ and under the heading sine^ or tangent^ &c., as required, the logarithm is found in a line with the angle. When the number of degrees is above 45°, and less than 90®, find the degrees and minutes in the right hand column headed angle^ and in the same line, above the title at the foot of the page, sine or tangent^ &c., find the logarithm in a line with the angle. When the number of degrees is between 90° and 180°, take their supple- ment to 180°; when between 180° and 270°, diminish them by 180°; and when between 270° and 360°, take their complement to 360°, and find the logarithm of the remainder as before. If the exact number of minutes is not found in the table, the logarithm of the nearest tabular angle is to be taken and increased or diminished as the case may be, by the due proportion of the difference of the logarithms of the angles greater and less than the given angle. Table No. X. — Rhumbs, or Points of the Compass. The Mariner's Compass is a circular card suspended horizontally, having a thin bar of steel magnetized, — the needle^ — for one of its diameters; the circumference of the card being divided into 32 equal parts, ox points^ and each point subdivided into quarters. A point of the compass is, therefore, equal to (360** ^ 32 = ) n° 15'. Table No. XL — Of Reciprocals of Numbers. The table No. XL contains the reciprocals of numbers from i to 1000. It has already been shown how to find the reciprocal of a number by means of logarithms. 38 MATHEMATICAL TABLES. TABLE No. L— LOGARITHMS OF NUMBERS FROM I TO 10,000. No. Log. 1^0. Log. No. Log. No. Log. 1 o.oooooo 26 1.414973 51 1.707570 76 1. 880814 2 0.301030 27 1.431364 52 1. 716003 77 1 1. 88649 1 3 0.477I2I 28 . 1.447158 53 1.724276 78 1.892095 4 0.602060 29 1.462398 54 1.732394 79 1.897627 5 0.698970 30 1 1.477121 55 1.740363 80 1.903090 6 0.778I5I 31 1.491362 56 I. 748188 81 1.908485 7 0.845098 32 1.505150 57 1-755875 82 1.913814 8 0.903090 33 1.518514 5« 1.76342S P 1. 919078 9 0954243 34 1.531479 59 1.770852 84 1.924279 lO I.OOOOOO 35 1.544068 60 1.77S151 85 1.929419 11 I.04I393 36 1-556303 61 1.785330 86 1.934498 12 I.079I8I 37 1.568202 62 1.792392 87 1.939519 13 1. 1 13943 38 1.579784 63 I. 799341 88 1.944483 H I.I46I28 39 1. 591065 64 1. 8061 80 89 I -949390 15 I.I7609I 40 1.602060 65 1.812913 90 91 1.954243 i6 1. 204120 41 1. 612784 66 1-819544 1.959041 'Z 1.230449 42 1.623249 ^7 1.826075 92 1.963788 i8 1-255273 43 1.633468 68 1.832509 93 1.968483 19 1.278754 44 1.643453 69 1.838849 94 1.973128 1 20 I. 301030 45 1. 653213 70 1.845098 1.851258 95 1.977724 21 1. 322219 46 1.662758 71 96 1. 98227 1 22 1.342423 47 1.672098 72 1.857332 97 1.986772 23 1. 361728 48 I.68i24< 73 1.863323 98 1. 991 226 24 1. 38021 1 49 1. 690196 74 1.869232 99 1.995635 25 I -397940 50 1.698970 75 I. 875061 100 2.000000 N I 2 3 4 5 — 1 6 7 8 9 D 432 100 00- 0000 0434 0868 1301 1734 2166 2598 3029 3461 3891 lOI 00- 4321 4751 51S1 5609 6038 6466 6894 7321 7748 8174 428 102 00- 8600 9026 9451 9876 425 102 01- 0300 0724 1 147 1570 1993 2415 424 103 01- 2837 3259 3680 4100 4521 4940 5360 5779 6197 6616 1 420 104 01- 7033 7451 7868 8284 8700 9116 9532 9947 1 417 104 02- 0361 0775 416 105 02- I I 89 1603 2016 2428 2841 3252 3664 4075 4486 4896 412 106 02- 5306 5715 6125 6533 6942 7350 7757 8164 8571 8978 408 107 02- 9384 9789 405 107 03- 0195 0600 1004 1408 1812 2216 2619 3021 404 108 03- 3424 3826 4227 4628 5029 5430 5830 6230 6629 7028 400 109 03- 7426 7825 8223 8620 9017 9414 981 1 398 109 N 04- 0207 0602 8 0998 9 397 I 2 3 4 5 6 7 D LOGARITHMS OF NUMBERS. 39 N I 1787 2 2182 3 2576 4 5 6 7 8 9 D 110 i 04- 1393 2969 3362 3755 4148 4540 iP^ 393 ! '" ' 04- 5323 5714 6105 6495 6885 7275 7664 8053 8442 8830 389 112 04- 9218 9606 9993 388 112 05- 0380 0766 1153 1538 1924 2309 2694 386 "3 05- 3078 3463 3846 4230 4613 4996 5378 5760 6142 6524 383 114 1 05- 6905 7286 7666 8046 8426 8805 9185 9563 9942 383 114 06- 0320 379 1 L15 06- 0698 1075 1452 1829 2206 2582 2958 3333 3709 4083 376 116 06- 4458 117 06- 8186 4832 5206 5580 5953 6326 6699 7071 7443 7815 373 8557 8927 9298 9668 • 380 "7 07- 0038 0407 0776 1 145 1514 370 118 07- 1882 2250 2617 2985 3352 3718 4085 4451 4816 5182 366 119 07- 5547 5912 6276 6640 7004 7368 773» 8094 8457 8819 363 120 1 07- 9181 9543 9904 • • * • • 362 120 08- 0266 0626 0987 1347 1707 2067 2426 360 121 08- 2785 3^44 3503 3861 4219 ^576 4934 5291 5647 6004 357 1 122 08- 6360 6716 7071 7426 7781 8136 8490 8845 9198 9552 355 : 123 08- 9905 355 123 09- 0258 0611 0963 '3'5 1667 2018 2370 2721 3071 353 1 124 09- 3422 3772 4122 4471 4820 5169 5518 5866 6215 6562 349 125 09- 6910 7257 7604 795 » 8298 8644 8990 9335 9681 348 125 10- 0026 346 126 10- 0371 0715 1059 1403 482! 1747 2091 2434 2777 3"9 3462 343 »27 xo- 3804 4146 4487 5169 5510 5851 6191 6531 6871 341 128 10- 7210 7549 7888 8227 8565 8903 9241 9579 9916 338 12S ] 11- 0253 337 129 XI- 0590 0926 1263 1599 1934 2270 2605 2940 3275 3609 335 130 XI- 3943 4277 461 1 4944 5278 8595 5611 5943 6276 6608 6940 333 131 IX- 7271 7603 7934 8265 8926' 9256 9586 9915 331 131 12- 0245 330 132 i 12- 0574 0903 4178 1231 1560 1888 2216 2544 5806 2871 3198 3525 328 m , "- 3852 4504 4830 5156 8399 5481 6131 6456 6781 325 134 12- 7105 7429 7753 8076 8722 9045 9368 9O90 323 1 *34 . 13- 0012 323 18« 13- 0334 o6« 0977 1298 1619 1939 2260 2580 2900 3219 321 '136 13- 3539 3858' 4177 4496 4814 5133 5451 5769 6086 6403 318 !i37 X3- 6721 7037 7354 7671 7987 8303 8618 8934 9249 9564 316 '^0 13- 9879 315 138 14- 0194 0508 0822 1 136 1450 1763 2076 2389 2702 3H 139 14- 3o»5 3327 3639 3951 4263 4574 4885 5196 5507 S8i8 3" 140 14- 6128 6438 6748 7058 7367 7676 7985 8294 8603 8911 309 141 14- 9219 9527 9835 308 :Hi ^5- •- • 0142 0449 0756 1063 1370 1676 1982 307 142 X5- 2288 2594 2900 3205 3510 lS'5 4120 4424 4728 5032 8061 305 143 15- 5336 5640 8664 5943 8965 6246 6549 6852 7154 7457 7759 303 144 15- 8362 9266 9567 9868 • • • • • • 302 144 16- 0168 0469 0769 1068 301 145 I 16- 1368 1667 1967 2266 2564 2863 3161 3460 3758 4055 299 M46 1 16- 4353 4650 4947 5244 5541 5838 6134 6430 9380 6726 7022 297 >47 i^ 7317 7613 7908 8203 8497 8792 9086 9674 9968 295 D N) I 2 3 4 5 6 7 8 9 40 MATHEMATICAL TABLES. N 148 I 2 3 4 5 6 7 8 9 D 293 17- 0262 0555 0848 II4I 1434 1726 2019 2311 2603 2895 149 17- 3186 3478 3769 4060 4351 4641 4932 5222 S5I2 5802 291 150 17- 6091 6381 6670 6959 7248 7536 7825 8113 8401 8689 289 151 17- 8977 9264 9552 9839 287 151 18- 0126 0413 0699 0986 1272 1558 287 IS2 z8- 1844 2129 2415 2700 2985 3270 3555 3839 4123 4407 ^?5 I S3 18- 4691 4975 5259 5542 5825 6108 6391 6674 6956 7239 283 154 i8~ 7521 7803 8084 8366 8647 8928 9209 9490 9771 281 »54 19- 0051 281 279 155 19- 0332 0612 0892 II7I 145 1 1730 2010 2289 2567 2846 156 19- 3125 3403 3681 3959 4237 45 »4 4792 5069 5346 5623 278 «57 19- 5900 6176 6453 6729 7«>5 7281 7556 7832 8107 8382 276 '55 19- 8657 8932 9206 9481 9755 275 158 2(>- 0029 0303 0577 0850 1124 274 159 20- 1397 1670 »943 2216 2488 2761 3033 3305 3577 3848 272 160 20> 4120 4391 4663 4934 5204 5475 8173 5746 E441 6016 6286 6556 271 161 20- 6826 7096 7365 7634 7904 8710 8979 9247 269 162 20- 9515 9783 268 162 21- c»5i 0319 0586 0853 1121 1388 1654 1921 267 163 21- 2188 2454 2720 2986 3252 3518 3783 4049 43H 4579 266 164 21- 4844 5109 5373 5638 5902 6166 6430 6694 6957 9585 7221 264 166 21- 7484 7747 8010 8273 8536 8798 9060 9323 9846 262 166 22- 0108 0370 0631 0892 "53 1414 1675 1936 2196 2456 261 167 22- 2716 2976 3236 3496 3755 4015 4274 4533 4792 5051 259 168 22- 5309 5568 5826 6084 6342 6600 6858 7115 7372 7630 258 169 22- 7887 8144 5400 8657 8913 9170 9426 9682 9938 257 169 23- 0193 256 170 23- 0449 0704 0960 1215 1470 1724 1979 2234 2488 2742 255 171 23- 2996 3250 3504 3757 401 1 4264 4517 4770 5023 5276 253 172 23- 5528 23- 8046 5781 S^33 6285 6537 6789 7041 7292 7544 7795 252 173 8297 8548 8799 9049 9299 9550 9800 251 173 24- • « B 0050 0300 250 174 24- 0549 0799 1048 1297 1546 179s 2044 2293 2541 2790 249 175 24- 3038 3286 3534 3782 4030 4277 4525 4772 5019 5266 248 176 24- 55^3 5759 8219 6006 6252 6499 6745 6991 72J7 7482 7728 246 177 24- 7973 8464 8709 8954 9198 9443 9687 9932 246 177 25- 0176 245 178 25- 0420 25- 2853 0664 o9oi; 3580 1395 1638 1881 2121 4548 2368 2610 243 179 3096 3338 3822 4064 4306 4790 5031 242 180 25- 5273 5514 5755 5996 6237 6477 6718 6958 7198 7439 241 181 25- 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 239 182 26- 0071 0310 0548 0787 1025 1263 1 501 1739 1976 2214 238 'S3 26- 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 237 184 26- 4818 5054 5290 5525 5761 5996 6232 6467 6702 6937 235 185 26- 7172 7406 7641 7875 81 10 8344 8578 8812 9046 9279 234 186 26- 9513 9746 9980 234 186 27- 0213 0446 0679 0912 1144 1377 1609 233 'SZ 27- 1842 2074 2306 2538 4850 2770 3001 3233 3464 3696 3927 232 188 27- 4158 4389 4620 5081 53" 5542 5772 6002 6232 230 D N I 2 3 4 5 6 7 8 9 LOGARITHMS OF NUMBERS. 41 N 1 o I 2 3 4 5 6 7 8 9 D 229 228 189 27- 6462 6692 6921 7151 7380 7609 7838 8067 8296 8525 190 ay- 8754 8982 921 1 9439 9667 9895 190 a8- a • • • « 0123 035' 0578 2849 0806 228 191 a8- 1033 1261 1488 1715 1942 2169 2396 2622 3075 227 192 aa- 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 226 193 a8- SSS7 i^6 6007 6232 6456 6681 6905 7130 7354 9589 7578 9812 225 194 a8- 7802 8249 8473 8696 8920 9143 9366 223 195 29- 0035 0257 Q480 0702 0925 1 147 1369 1591 3804 1813 2034 222 196 29- 2256 2478 2699 2920 3141 3363 3584 4025 4246 221 '^Z ag- 4466 4687 4907 5127 5347 5567 5787 6007 6226 6446 220 198 39- 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 219 X99 29-8853 9071 9289 9507 9725 9943 218 '99 30- 0161 0378 0595 0813 2x8 200 30- 1030 1247 1464 1681 180R 2114 233« 2547 2764 2980 217 201 30- 3196 3412 3628 3844 4059 4275 4491 4706 4921 5136 216 202 30- 535 > 5566 5781 5996 8137 621 1 6425 6639 6854 7068 7282 215 203 30- 7496 7710 7924 8351 8564 8778 8991 9204 9417 213 204 30- 9630 9843 • a • • • « • 213 204 3X- 0056 0268 0481 0693 0906 1118 1330 1542 212 1205 31- 1754 1966 2177 2389 2600 2812 3023 3234 3445 3656 211 ! 206 31- 3867 4078 4289 4499 4710 4920 5130 5340 5551 5760 210 207 31- 5970 31- 8063 6180 6390 6599 6809 7018 7227 7436 7646 7854 209 208 8272 8481 8689 8898 9106 9314 9522 9730 9938 208 209 210 32- 0146 0354 0562 0769 0977 1 184 1391 1598 1805 2012 207 32- 2219 2426 2633 2839 3046 3252 3458 3665 3871 4077 206 211 32- 4282 44^ ;8 4694 4899 5105 5310 5516 5721 5926 6131 205 212 32- 6336 32- 8380 6541 6745 6950 7155 7359 7563 7767 7972 8176 204 213 8583 8787 8991 9194 9398 9601 9805 204 2J3 33- 0008 0211 203 2H 33- 0414 0617 G819 1022 1225 1427 1630 1832 2034 2236 202 215 33- 2438 2640 2842 3044 3246 3447 3649 3850 4051 4253 202 216 33- 4454 4655 4856 5057 5257 5458 5658 5859 6059 6260 201 217 33- 6460 6660 6860 7060 7260 7459 7659 7858 «os« 8257 200 m 218 33- 8456 8656 8855 9054 9253 9451 9650 9849 200 218 34- • • • 0047 0246 199 219 34- 0444 0642 0841 1039 1237 1435 1632 1830 2028 2225 198 220 34- 2423 2620 2817 3014 3212 3409 3606 3802 3999 4196 197 . 221 34- 4392 4589 4785 4981 5178 5374 5570 5766 5962 6157 196 222 34- 6353 6549 6744 6939 7135 7330 7525 7720 7915 8110 195 223 34- 8305 8500 8694 8889 9083 9278 9472 9666 9860 194 223 35- « • • 0054 1989 194 224 35- Q248 0442 c6i6 0829 1023 1216 1410 1603 1796 193 '225 35- 21^3 2375 2568 2761 4876 3147 3339 3532 3724 3916 193 1 226 35- 4108 4301 4493 6408 4685 5068 5260 5452 5643 5834 192 227 35- 6026 6217 6599 6790 6981 7172 7363 7554 7744 191 1 228 35- 7935 8125 8316 8506 86q6 8886 9076 9266 9456 9646 190 1 229 35-9835 189 1 1 229 35- 0025 0215 0404 0593 0783 0972 1 161 1350 1539 189 D N I 2 3 4 5 6 7 8 9 i 42 MATHEMATICAL TABLES. N 6 8 D 230 36- 1728 191 7 2105 2294 2482 231 36-3612 3800 3988 4176 4363 232 36- 5488 5675 5862 6049 6236 233 36- 7356 7542 7729 7915 8101 234 36- 9216 9401 9587 9772 9958 2671 2859 3048 4551 4739 4926 6423 6610 6796 8287 8473 8659 3236 3424 5"3 5301 6983 7169 8S45 9030 P • a • [88 [88 187 [86 [86 234 235 236 237 238 239 239 37- 37- 1068 37- 2912 37- 4748 37- 6577 37- 8398 38- 1253 1437 1622 1806 3096 3280 3464 3647 4932 5"5 5298 5481 6759 6942 7124 7306 8580 8761 8943 9124 0143 1991 3831 5664 7488 9306 0328 2175 4015 5846 7670 9487 05 '3 2360 4198 6029 7852 9668 0698 2544 4382 6212 8034 9849 0883 2728 4565 6394 8216 0030 (85 [84 [84 ^83 [S2 [82 [81 240 241 242 243 244 38- 021 I 38- 2017 38- 3815 38- 5606 38- 7390 0392 2197 3995 578§ 7568 0573 2377 4174 5964 7746 0754 2557 4353 6142 7923 0934 2737 4533 6321 8101 I I 15 1296 1476 2917 3097 3277 4712 4891 5070 6499 6677 6856 8279 8456 8634 1656 1837 181 3456 3636 180 5249 5428 179 7034 7212 178 8S11 8989 178 245 245 246 247 248 249 38- 9166 39- 39- 0935 39- 2697 39- 4452 39- 6199 9343 9520 9698 9875 1112 2873 4627 6374 1288 3048 4802 6548 1464 3224 4977 6722 1641 3400 5152 6896 0051 1817 3575 5326 7071 0228 1993 3751 5501 724s 0405 2169 3926 5676 7419 0582 2345 4101 5850 7592 0759 2521 4277 6025 7766 77 77 76 76 75 74 250 251 251 252 253 254 255 256 257 257 258 259 39- 7940 39- 9674 40- 40- 1401 40- 3121 40- 4834 8114 9847 1573 3292 5cx>s 8287 8461 8634 0020 1745 3464 S176 0192 1917 3635 5346 0365 2089 3807 5517 8808 8981 9154 9328 9501 0538 2261 3978 $688 071 1 2433 4149 5858 0883 2605 4320 6029 1056 2777 4492 6199 1228 2949 4663 6370 73 73 73 72 71 71 40- 6540 40- 8240 40- 9933 41- 41- 1620 41- 3300 6710 8410 0102 1788 3467 6881 8579 0271 1956 3635 7051 8749 • • • • • 0440 2124 3803 7221 8918 0609 2293 3970 739X 9087 0777 2461 4137 7561 9257 • • 0946 2629 4305 7731 9426 1114 2796 4472 7901 9595 1283 2964 4639 8070 9764 1451 3>32 4806 70 [69 [69 [69 [68 67 260 261 262 263 263 264 41- 4973 41- 6641 41- 8301 41- 9956 42- 42- 1604 5140 6807 8467 0121 1768 5307 6973 8633 0286 1933 5474 7139 8798 0451 2097 5641 ] 5808 7306 8964 0616 2261 7472 9129 0781 2426 5974 7638 9295 0945 2590 6141 7804 9460 1 110 2754 6308 7970 9625 1275 2915 6474 8135 9791 1439 3082 [67 [66 [6s [65 [65 [64 265 266 267 268 269 269 270 271 N 42- 3246 42- 4882 42- 651 I 42- 8135 42- 9752 43- 3410 5045 6674 8297 9914 3574 5208 6836 8459 3737 5371 6999 8621 3901 5534 7161 8783 0075 0236 0398 4065 5697 7324 8944 4228 5860 7486 9106 4392 6023 7648 9268 4555 6186 781 1 9429 4718 6349 7973 9591 0559 0720 0881 1042 1203 43- 1364 43- 2969 1525 3130 1685 3290 1846 3450 2007 3610 2167 3770 2328 3930 2488 4090 7 2649 4249 8 2809 4409 [64 163 t62 [62 [62 [61 [61 60 D LOGARITHMS OF NUMBERS. 43 N I 2 3 4 5 6 7 8 9 D 272 43- 45^ 4729 4888 5048 5207 5367 5526 5685 5844 6004 VS9 273 43- 6163 6322 64S1 6640 6799 6957 7116 7275 7433 7592 159 274 43- 7751 7909 8067 8226 8384 8542 8701 8859 9017 9175 158 275 43- 9333 9491 9648 9806 9964 158 275 44- 0122 0279 0437 0594 0752 158 276 44- 09C39 1066 1224 1381 iS3^ 1695 1852 2009 2166 2323 157 277 44- 2480 2637 2793 2950 3106 3263 3419 3576 3732 3889 157 278 44- 4045 4201 4357 4513 4669 4825 4981 5137 5293 5449 156 279 44- 5604 5760 5915 6071 6226 6382 6537 6692 6848 7003 155 280 44- 7158 Z3'3 7468 7623 7778 7933 8088 8242 8397 8552 155 281 44- 8706 8S61 9015 9170 9324 9478- 9633 9787 9941 154 281 45- 0095 154 282 45- 0249 0403 0557 071 1 0865 1018 1 172 1326 1479 1633 154 ^S^ 45- 1786 1940 2093 2247 2400 2553 2706 2859 3012 3165 153 284 45- 3318 3471 3624 3777 3930 4082 4235 4387 4540 4692 153 285 45- 4845 4997 5150 5302 5454 5606 5758 5910 6062 6214 ^52 286 45- 6366 6518 6670 6821 6973 7125 7276 7428 7579 7731 152 287 45- 7882 «033 8184 8336 8487 8638 8789 8940 9091 9242 151 288 45- 9392 9543 9694 9845 9995 151 2SS 46- ...... 0146 0296 0447 0597 0748 151 289 46- 0S98 1048 1198 1348 1499 1649 1799 1948 2098 2248 150 280 46- 2398 2548 2697 2847 2997 3146 3296 3445 3594 3744 150 291 46- 3893 4042 4191 4340 4490 4639 4788 4936 5085 5234 149 292 46- 5383 46- 6868 5532 5680 5829 5977 6126 6274 6423 6571 6719 149 293 7016 7164 7312 7460 7608 7756 7904 8052 8200 148 294 46- 8347 8495 9969 8643 8790 8938 9085 9233 9380 9527 9675 148 295 45- 9822 • B • 147 295 47- 01 16 0263 0410 0557 0704 0851 0998 1 145 147 296 47- 1292 1438 I5«5 1732 1878 2025 2171 2318 2464 2610 146 297 47- 2756 2903 3049 3195 3341 3487 3633 3779 3925 4071 146 298 47- 4216 4362 4S08 4653 4799 4944 5090 5235 5381 5526 146 299 47- 5671 5816 5962 6107 6252 6397 6542 6687 6832 6976 145 300 47- 7121 7266 741 1 7555 7700 7844 7989 8133 8278 8422 145 301 47- 8566 871 1 8855 8999 9143 9287 9431 9575 9719 9863 144 3«>2 48— 0007 0151 1586 0294 0438 0582 0725 0869 1012 "§S 1299 144 393 48- 1443 1729 1872 2016 2K9 2302 2445 2588 2731 143 304 48- 2874 3016 3^59 3302 3445 3587 3730 3872 4015 4157 M3 305 48- 4300 4442 4585 4727 4869 5011 5153 5295 5437 5579 142 306 43- 5721 5863 6005 6147 6289 6430 6572 6714 6855 6997 142 3^ 48- 7138 7280 7421 7563 7704 7845 7986 8127 8269 8410 141 SoS 48- 8551 8692 8833 8974 9114 9255 9396 9537 9677 9818 141 3P9 48- 9958 140 . 3P9 49- 0099 0239 0380 0520 0661 0801 0941 108 1 1222 140 310 49- 1362 1502 1642 1782 1922 2062 2201 2341 2481 2621 140 3^1 49- 2760 29CX> 3040 3179 3319 3458 3597 3737 3876 4015 139 312 49- 4155 4294 4433 4572 471 1 4850 4989 5128 5267 5406 139 313 49- 5544 5683 5822 5960 6099 6238 6376 6515 6653 6791 139 ' 3«4 49- 6930 7068 7206 7344 7483 7621 7759 7897 8035 8173 138 N I 2 3 4 5 6 7 8 9 D 44 MATHEMATICAL TABLES. N 315 316 316 317 318 319 49- 83" 49- 9687 50- 1059 50- 2427 50- 3791 8448 9824 1 196 2564 3927 8586 9962 1333 2700 4063 8724 8862 • •• • 0099 1470 2837 4199 8 8999 9137 9275 9412 9550 0236 0374 1607 1744 2973 4335 3109 4471 0511 1880 3246 4607 0648 2017 3382 4743 0785 2154 35^8 4878 0922 2291 36SS 5014 D 138 ^37 t37 [37 ^36 ^36 320 321 322 323 323 324 50- 5150 50- 6505 50- 7856 50- 9203 51- 51- 0545 5286 6640 7991 9337 5421 6776 8126 9471 5557 691 1 8260 9606 5693 7046 8395 9740 0679 0813 0947 1081 5828 7181 8530 9874 1215 5964 7316 8664 0009 1349 6099 7451 8799 0143 1482 6234 7586 8934 0277 1616 6370 7721 9068 041 1 1750 ^36 135 ^35 34 '34 134 325 326 327 328 329 ' 51- i88j 51- 3218 51- 4548 51- 5874 51- 7196 2017 3351 4681 6006 7328 2151 3484 4813 6139 7460 2284 3617 4946 6271 7592 2418 3750 5079 6403 7724 2551 3883 521 1 6535 785s 2684 4016 5344 6668 7987 2818 4149 5476 6800 81 19 2951 4282 5609 6932 8251 3084 4415 5741 7064 8382 ^33 '33 133 [32 '32 380 331 331 332 333 334 51- 8514 51- 9828 52- 52- 1138 52- 2444 52- 3746 8646 9959 1269 2575 3876 8777 8909 9040 0090 1400 2705 4006 022 X IJ30 2835 4136 0353 1661 2966 4266 9171 9303 9434 9566 9697 0484 1792 3096 4396 0615 1922 3226 4526 0745 2053 3356 4656 0876 2183 3486 4785 1007 2314 3616 4915 '31 31 '31 31 [30 '30 385 336 337 338 338 339 52- 5045 52- 6339 52- 7630 52- 8917 53- 53- 0200 5174 6469 7759 9045 5304 6598 7888 9174 5434 6727 8016 9302 5563 6856 8145 9430 5693 6985 8274 9559 5822 7114 8402 9687 5951 7243 8531 9815 6081 7372 8660 9943 0328 0456 0584 0712 0840 0968 1096 1223 6210 7501 8788 0072 1351 [29 [29 129 [28 [28 28 840 341 342 343 344 53- 1479 53- 2754 53- 4026 53- 5294 53- 6558 1607 2882 4153 5421 6685 1734 3009 4280 5547 6811 1862 3136 4407 5674 6937 1990 3264 4534 5800 7063 2117 3391 4661 5927 7189 2245 3518 4787 6053 7315 2372 3645 4914 6180 7441 2500 3772 5041 6306 7567 2627 3899 5167 6432 7693 [28 [27 [27 [26 [26 346 346 346 347 348 349 53- 7819 53- 9076 54- 54- 0329 54- 1579 54- 2825 7945 9202 0455 1704 2950 8071 9327 0580 1829 3074 8197 9452 0705 1953 3199 8322 9578 0830 2078 3323 8448 9703 • • • • 0955 2203 3447 8574 9829 1080 2327 3571 8699 9954 1205 2452 3696 8825 8951 0079 1330 2^76 3820 0204 1454 2701 3944 [26 [26 [25 [25 25 124 360 351 352 353 354 354 365 356 357 N 54- 4068 54- 5307 54- 6543 54- 7775 54- 9003 55- 4192 5431 6666 7898 9126 4316 5555 6789 8021 9249 4440 5678 6913 8144 9371 4564 5802 7036 8267 9494 4688 5925 7159 8389 9616 4812 6049 7282 8512 9739 4936 6172 7405 863s 9861 5060 6296 7529 8758 9984 5183 6419 7652 8881 0106 55- 0228 0351 0473 0595 0717 55- '450 1572" 1694 1816 1938 55- 2668 2790 29 I I 3033 3155 0840 2060 3276 0962 2181 3398 1084 2303 3519 1206 2425 3640 8 1328 2547 3762 [24 124 '23 23 23 '23 [22 [22 [21 D LOGARITHMS OF NUMBERS. 45 N I 2 3 4 5 6 7 8 9 D 121 358 55- 3883 4004 4126 4247 4368 4489 4610 4731 4852 4973 359 55- 5094 52 » 5 5336 5457 5578 5699 5820 5940 6061 6182 121 360 55- 6303 6423 6544 6664 6785 6905 8108 7026 7146 7267 7387 120 361 55- 7507 7627 7748 7868 7988 8228 8349 8469 8589 120 362 1 55- 8709 8829 8948 go68 91^ 9308 9428 9548 9667 9787 120 363 55- 9907 120 363 56- 0026 0146 0265 0385 0504 1698 0624 0743 0863 0982 "9 364 56- iioi 1221 1340 H59 1578 1817 1936 2055 2174 119 365 56- 2293 2412 2531 2650 2769 2887 3006 3125 3244 3362 119 366 56- 3481 3600 37»8 3837 3955 4074 4192 43" 4429 4548 119 367 56- 4666 4784 4903 5021 5139 5257 5376 5494 5612 5730 118 368 56- 5848 5966 6084 6202 6320 6437 6555 6673 6791 6909 118 369 56- 7026 7144 7262 7379 7497 7614 7732 7849 7967 8084 118 370 56- 8202 8319 8436 8554 8671 8788 8905 9023 9140 9257 117 371 55- 9374 9491 9608 9725 9842 9959 ^ 117 37 « 57- -]•.. 0076 0193 0309 0426 "7 372 57- 0543 0660 0776 0893 2058 lOIO 1 126 1243 1359 1476 1592 117 373 57- 1709 1825 1942 2174 2291 2407 2523 2639 2755 116 374 57- 2872 2988 3»04 3220 3336 3452 3568 3684 3800 3915 116 375 57- 403' 4147 4263 4379 4494 4610 4726 4841 4957 5072 116 376 57- 5188 5303 S419 ^^ 5650 5765 5880 5996 61 1 1 6226 "5 377 57- 6341 6457 6572 6802 6917 7032 7147 8295 7262 7377 "5 378 57- 7492 7607 7722 7836 7951 8066 81S1 8410 8525 "5 379 57- 8639 8754 8868 8983 9097 9212 9326 9441 9555 9669 114 380 57- 9784 9898 ■• «■■ •«■••■ "4 380 58- 0012 0126 0241 0355 0469 0583 0697 081 I 114 381 58- 092s 1039 "53 1267 1381 1495 1608 1722 1836 1950 2972 3085 "4 382 58- 2063 2177 2291 2404 2518 2631 2745 2858 114 383 58- 3199 33»2 3426 3539 3652 3Z^§ 3879 3992 4105 4218 "3 ^ 58- 4331 444* 4557 4670 4783 4896 5009 5122 5235 5348 "3 385 58- 5461 5574 5686 5799 5912 6024 6137 6250 6362 6475 "3 386 58- 6587 6700 6812 6925 7037 7149 P^^ 7374 7486 7599 112 387 58- 77" 7823 7935 8047 8160 8272 8384 8496 8608 8720 112 38S 58- 8832 8944 9056 9167 9279 9391 9503 9615 9726 9838 112 . 389 58- 9950 112 389 59- 0061 0173 0284 0396 0507 0619 0730 0842 0953 112 300 59- 1065 1 1 76 1287 1399 15 10 I62I 1732 1843 1955 2066 III 391 59- 2177 2288 2399 2510 2621 2732 2843 2954 3064 3175 III 392 59- 3286 3397 3«>o8 3618 3729 3840 3950 4061 41 7 I 4282 III 393 59- 4393 4503 4614 4724 4834 4945 5055 5165 5276 5386 110 394 59- 5496 . 5606 5717 5827 5937 6047 6157 6267 6377 6487 no 395 59- 6597 6707 6817 6927 7037 7146 7256 7366 7476 7586 no 396 59- 7695 7805 59- 879* 8900 7914 8024 8134 8243 8353 8462 8572 8681 no 397 9009 9119 9228 9337 9446 9556 9665 9774 109 398 59- 9883 9992 109 398 60- OIOI 0210 0319 I 0428 0537 0646 0755 ^864 109 399 400 60- 0973 1082 II9I 1299 1408 ^ 1517 1625 1734 1843 1951 109 60- 2060 2169 2277 2386 2494 2603 5 271 1 6 2819 7 2928 3036 8 9 108 N 1 I 2 3 4 46 MATHEMATICAL TABLES. N I 2 3 4 5 6 7 8 9 D 108 401 60- 3144 3253 3361 3469 3577 3686 3794 3902 4010 41 18 402 60- 4226 4334 4442 ^§50 4658 4766 4874 4982 5089 5197 108 403 60- 5305 5413 5521 5628 5736 5844 595' 6059 6166 6274 108 404 60- 6381 6489 6596 6704 6811 6919 7026 7133 7241 7348 107 405 60- 7455 7562 7669 SJ? 7884 \ 7991 8098 8205 8312 8419 107 406 60- 8526 8633 8740 8954 ; 9061 9167 9274 9381 9488 107 407 60- 9594 9701 9808 9914 107 407 61- 0021 0128 0234 0341 0447 0554 107 408 6x- 0660 0767 0^73 0979 1086 1 192 1298 1405 1511 1617 106 409 61- 1723 1829 1936 2042 2148 2254 2360 2466 2572 2678 X06 410 61- 2784 2890 2996 3102 3207 3313 3419 3525 3630 4686 3736 106 411 61- 3842 3947 4053 4159 4264 4370 4475 4581 4792 106 412 61 - 4897 5003 5108 5213 5319 5424 5529 5634 5740 5845 105 413 61- 5950 6055 6160 6265 6370 6476 6581 6686 6790 6895 105 414 61- 7000 7105 7210 7315 7420 7525 7629 7734 7839 7943 105 415 61- 8048 8153 9198 8257 8362 8466 8571 8676 8780 8884 8989 105 416 61-9093 9302 9406 9511 9615 9719 9824 9928 105 4x6 6a- «■•••• 0032 104 417 62- 0136 0240 1280 0344 0448 0552 0656 0760 0864 0968 1072 104 418 62- 1 1 76 1384 1488 1592 1695 1799 1903 2007 21 10 104 419 62- 2214 2318 2421 2525 2628 2732 2835 2939 3042 3146 104 420 62- 3249 3353 4488 3559 3663 3766 3869 3973 4076 4179 103 421 62- 4282 4385 4591 4695 4798 4901 5004 5107 5210 103 422 6a- 5312 5415 5518 5621 5724 5827 5929 6032 6135 6238 »03 423 62- 6340 6443 6546 6648 6751 ^^53 6956 7980 Z°5^ 7161 P^^ 103 424 62- 7366 7468 7571 7673 7775 7878 8082 8185 8287 102 425 62- 8389 8491 8593 8695 8797 8900 9002 9104 9206 9308 102 426 62- 9410 9512 9613 9715 9817 9919 102 426 63- 0021 0123 0224 0326 102 427 63- 0428 0530 0631 0733 ^35 0936 1038 1 139 1241 1342 102 428 63- 1444 1545 1647 1748 1849 1951 2052 2153 2255 2356 lOI 429 63- 2457 2559 2660 2761 2862 2963 3064 3«65 3266 3367 lOI 480 63- 3468 3569 3670 3771 3872 3973 4074 4175 4276 4376 lOI 431 63- 4477 4578 4679 4779 4880 4981 5081 5182 5283 5383 lOI 432 63- 5484 5584 5685 5785 5886 5986 6087 6ih7 6287 6388 100 433 63- 6488 6588 6688 6789 6889 6989 7089 7189 7290 7390 8389 100 434 63- 7490 7590 7690 7790 7890 7990 8090 8190 8290 100 485 63- 8489 8589 8689 8789 8888 8988 9088 9188 9287 9387 100 436 63- 9486 9586 9686 9785 9885 9984 lOO 436 64- 0084 0183 0283 0382 99 437 64- 0481 0581 0680 0779 0879 0978 1077 "77 1276 1375 99 438 64- 1474 1573 1672 1771 1871 1970 2069 2168 2267 2366 99 439 64- 2465 2563 2662 2761 2860 2959 3946 3058 4044 3156 4143 3255 4242 3354 4340 99 99 440 64- 3453 3551 3650 3749 3847 441 64- 4439 4537 4636 4734 4832 4931 5029 5127 5226 5324 98 442 64- 5422 5521 5619 5717 5815 5913 601 1 6110 6208 6306 98 443 64- 6404 6502 6600 6698 6796 6894 6992 7089 7187 7285 98 444 N 64- 7383 7481 7579 7676 7774 7872 7969 8067 8165 8262 98 I 2 3 4 s 6 7 8 9 D LOGARITHMS OF NUMBERS. 47 N o P I 2 3 4 S 6 7 8 9 D 445 64- 8360 8458 8555 8653 8750 8848 8945 9043 9140 9237 97 446 64- 9335 9432 9530 9627 9724 9S21 9919 97 446 65- 0016 0113 0210 97 447 65- 0308 0405 0502 0599 0696 0793 0890 0987 1084 1181 97 i 44S 65- 1278 1375 1472 1569 1666 1762 'S^? 1956 2053 2150 97 449 65- 2246 2343 2440 2536 2633 2730 2826 2923 3019 311^ 97 450 65- 3213 3309 3405 3502 3598 3695 3791 3888 3984 4080 96 4S» 65- 4177 4273 4369 4465 4562 4658 4754 4850 4946 5042 96 452 65- 5138 5235 5331 5427 5523 5619 5715 5810 5906 6002 96 453 65- 6098 6194 6290 6386 6482 6577 6673 6769 6864 6960 96 454 65- 7056 7152 7247 7343 7438 7534 7629 7725 7820 7916 96 455 65- Soil 8107 8202 8298 8393 8488 8584 8679 8774 8870 95 456 65- 8965 9060 9155 9250 9346 9441 9536 963^ 9726 9821 95 457 65- 9916 ••■•»■ 95 457 66- OOII 0106 0201 0296 0391 0486 0581 0676 0771 95 45^ 66- 0865 0960 1055 1 150 1245 1339 1434 2380 1529 1623 1718 95 459 66- 1813 1907 2002 2096 2191 2286 2475 2569 2663 95 460 65- 2758 2852 2947 3041 3135 3230 3324 3418 3512 3607 94 461 66- 3701 3795 3889 3983 4078 4172 4266 4360 4454 4548 94 462 66- 4642 4736 4830 4924 5018 5112 5206 5299 5393 5487 94 463 66- 5581 5675 5769 5862 5956 6050 6143 6237 6331 6424 94 464 66- 6518 6612 6705 6799 6892 6986 7079 7173 7266 7360 94 465 66- 7453 7546 7640 7733 7826 7920 8013 8106 8199 8293 93 466 66- 8386 8479 8572 8665 8759 88^2 8945 9038 9J31 9224 93 467 66- 9317 9410 9503 9596 9689 9782 9875 9967 93 467 67- • • • a • 0060 0153 1080 93 46S 67- 0246 0339 0431 0524 0617 0710 0802 089s 0988 93 469 67- "73 1265 1358 145 I 1543 1636 1728 1821 1913 2005 93 470 67- 2098 2190 2283 2375 2467 2560 2652 2744 2836 2929 92 471 67- 3021 3"3 3205 3297 3390 3482 3574 3666 3758 3850 92 472 67- 3942 4034 4126 4218 4310 4402 4494 4586 4677 4769 92 473 67- 4861 4953 5045 5137 5228 5320 5412 5503 5595 5687 92 474 67- 5778 5870 5962 6053 6145 6236 6328 6419 6511 6602 92 475 67- 6694 6785 7698 6876 6968 7059 7151 7242 8154 7333 7424 8336 7516 91 476 67- 7607 67- 8518 7789 7881 7972 8063 8245 8427 91 477 8609 8700 8791 8882 ^Z3 9064 9155 9246 9337 91 478 67- 9428 9519 9610 9700 9791 9882 9973 91 478 68- C063 0154 0245 91 479 68- 0336 0426 0517 0607 0698 0789 0879 0970 1060 1151 91 480 68- 1241 1332 1422 I5I3 2416 1603 1693 1784 1874 1964 2055 90 481 68- 2145 2235 2326 2506 2596 2686 2777 2867 2957 90 4S2 68- 3047 3137 3227 3317 3407 3497 3557 3^77 3767 3857 90 : 483 68- 3947 4037 4127 4217 4307 4396 4486 4576 4666 4756 90 484 68- 4845 4935 5025 5II4 5204 5294 5383 5473 5563 5652 90 485 68- 5742 5831 5921 6010 61CX) 6189 6279 6368 6458 6547 89 486 68- 6636 6726 6815 6904 6994 7083 7172 7261 7351 7440 89 487 68- 7529 7618 7707 7796 7886 7975 8064 8153 8242 8331 89 488 68- S420 8509 8598 8687 8776 8865 8953 9042 9131 9220 89 1489 68- 9309 9398 9486 9575 9664 9753 5 9841 9930 89 D j N } I 2 3 4 6 7 8 9 48 MATHEMATICAL TABLES. N 4«9 1490 491 492 493 494 69- 8 0019 0107 69- 0196 69- 1081 69- 1965 69- 2847 69-3727 0285 1 170 2053 2935 381S 0373 1258 2142 3023 3903 0462 1347 2230 3111 3991 0550 1435 2318 3>99 4078 0639 1524 2406 3287 4166 0728 1612 2494 3375 4254 0816 1700 2583 3463 4342 0905 17S9 2671 3551 4430 0993 1877 2759 3639 4517 D 89 89 88 88 88 88 496 496 497 498 499 69- 4605 69- 5482 69- 6356 69- 7229 69- 8101 4693 5569 6444 7317 8188 4781 5657 6531 7404 8275 4868 5744 6618 7491 8362 4956 5832 6706 7578 8449 5044 5919 6793 7665 8535 513" 6007 6880 7752 8622 5219 6094 6968 7839 8709 5307 6182 7055 7926 8796 5394 6269 7142 8014 8883 88 87 87 87 87 500 501 502 503 504 69- 8970 69- 9838 70- 70- 0704 70- 1568 70- 2431 9057 9924 0790 1654 2517 9144 9231 9317 001 1 0877 1741 2603 0098 0963 1827 2689 0184 1650 1913 2775 9404 9491 9578 9664 9751 0271 1 136 1999 2861 0358 1222 2086 2947 0444 1309 2172 3033 0531 1395 2258 3"9 0617 1482 2344 3205 87 87 87 86 86 86 505 506 507 508 509 510 511 512 512 5*3 514 70- 3291 70- 4151 70- 5008 70- 5864 70- 6718 3377 4236 5094 5949 6803 3463 4322 5179 60- 6^ 3549 4408 5265 6120 6974 3635 4494 5350 6206 7059 3721 4579 5436 6291 7144 3807 4665 5522 6376 7229 3893 4751 5607 6462 7315 3979 4837 5693 6547 7400 4065 4922 5778 6632 7485 70- 7570 70- 8421 70- 9270 71- 71- 0117 71- 0963 7655 8506 9355 0202 1048 7740 8591 9440 0287 1 132 7826 8676 9524 0371 1217 791 1 8761 9609 0456 1301 7996 8846 9694 0540 1385 8081 8931 9779 0625 1470 8166 9015 9S63 0710 1554 8251 9100 9948 0794 1639 8356 9185 • • • • • 0033 0879 1723 86 86 86 85 85 f5 85 85 84 516 516 518 5«9 71- 1807 71- 2650 71- 3491 71- 4330 71- 5*67 1892 2734 3575 4414 5251 1976 2818 3659 4497 5335 2060 2902 3742 4581 5418 2144 2986 3826 4665 5502 2229 3070 3910 4749 5586 2313 3>54 3994 4833 5669 2397 3238 4078 4916 5753 2481 3323 4162 5000 5836 2566 3407 4246 5084 5920 84 84 84 84 84 |520 , 521 522 i 523 524 524 71- 6003 71- 6838 71- 7671 71- 8502 71- 9331 72- 6087 6921 7754 8585 9414 6170 7004 7837 8668 9497 6254 7088 7920 8751 9580 6337 7171 8003 8834 9663 6421 7254 8086 8917 9745 6s(H 7338 8169 9000 9828 6588 7421 8253 9083 9911 6671 7504 8336 9165 9994 6754 7587 8419 9248 0077 7a- 01 g9 0242 0325 0407 0490 72- 0986 1068 I 151 1233 1316 7a- i8u 1893 1975 2058 2140 7a- 2634 2716 2798 2881 2963 72- 3456 3538 3620 3702 3784 0573 0655 0738 0821 0903 1398 1481 1563 1646 1728 2222 2305 2387 2469 2552 3045 3127 3209 3291 3374 3866 3948 4030 41 12 4194 72- 4276 4358 4440 4522 4604 72- 5095 5176 5258 5340 5422 72- 5912 5993 6075 61 s6 6238 7a- 6727 0^ 6890 6972 7053 4685 4767 4849 4931 50'3 5503 5585 5667 5748 5830 6320 6401 64S3 6564 6646 7134 7216 7297 7379 7460 «3 83 P 83 83 82 82 82 82 82 82 82 81 8 D LOGARITHMS OF NUMBERS. 49 1 1 1« N 534 I 2 3 4 5 6 7 8 9 D 81 81 72- 7541 7623 7704 7785 7866 7948 8029 8110 8191 8273 9084 »d5 1 7^ 8354 8435 8516 8597 8678 8759 8841 8922 9003 536 72- 9165 9246 9327 9408 9489 9570 9651 9732 9813 9893 81 537 72- 9974 ^ 81 537 73- OOS5 0136 0217 0298 0378 0459 0540 0621 0702 81 1 538 1 73- 0782 086^ 0944 1024 1105 1186 1266 1347 1428 1508 81 .{ 539 73- 1589 1669 1750 1830 1911 1991 2072 2152 2233 2313 81 HO 73- 2394 2474 2S55 2635 3438 2715 2796 2876 2956 3037 3"7 80 541 73- 3>97 3278 3358 3518 3598 3679 3759 3839 3919 80 542 73- 3999 4079 4160 4240 4320 4400 4480 4560 4640 4720 80 543 73- 4800 4880 4960 5040 5120 5200 5279 5359 5439 55>9 80 I 544 73- 5599 5679 5759 5838 5918 5998 6078 6157 6237 6317 io 546 73- 6397 6476 6556 6635 6715 6795 6874 6954 7034 7113 80 546 73- 7193 7272 7352 7431 751 1 7590 7670 7749 7829 7908 79 547 73- 7987 8067 8x46 8225 8305 8384 8463 8543 8622 8701 79 548 73- 8781 8860 8939 9018 9^7 9'?Z 9256 9335 9414 9493 79 549 73- 9572 9651 9731 9810 9889 9968 79 549 74- 0047 0126 0205 0284 79 i 650 74- 0363 0442 0521 0600 0678 0757 0836 0915 0994 '2?3 79 .'551 74- 1152 1230 1309 1388 1467 1546 1624 1703 1782 i860 79 552 74- 1939 2018 2096 2175 2254 2332 241 1 2489 2568 2647 79 553 74- 2725 2804 2882 2961 3039 3II8 3196 3275 3353 3431 78 554 74- 35>o 3588 3667 3745 3823 3902 3980 4058 4136 4215 78 n 555 1 74- 4293 4371 4449 4528 4606 4684 4762 4840 4919 4997 75 556 74- 5075 5153 5231 5309 5387 S465 5543 5621 5699 5777 78 557 74- 5855 5933 601 1 6089 6167 6245 6323 6401 6479 6556 78 558 74- 6634 6712 6790 6868 6945 7023 7101 7179 7256 7334 78 559 74- 7412 7489 7567 7645 7722 7800 7878 7955 8033 8110 78 560 74 81S8 8266 8343 8421 8498 8576 8653 8731 8808 8885 77 ;50i 74-8963 9040 9118 9195 9272 9350 9427 9504 9582 9659 77 562 i 74- 9736 Q814 9891 77 562 ; 75- 0045 0123 0200 0277 0354 0431 77 N 563 75- 0508 0586 ii663 0740 0817 0894 0971 1048 1 125 1202 77 ■564 75- 1279 1356 1433 1510 1587 1664 1 741 1818 1895 1972 77 566 75- 2048 2125 2202 2279 2356 2433 2509 2586 2663 2740 77 >66 75- 2816 2893 2970 3047 3123 3200 3277 3353 3430 3506 77 '^ 75- 3583 S66o 3736 3813 3889 3966 4042 41 19 4195 4272 77 568 75- 4348 4425 4501 4578 4654 4730 4807 4883 4960 5036 76 S^ 75- 5"2 5189 5265 5341 5417 5494 5570 5646 5722 5799 76 570 75- 5875 595 « 6027 6103 6180 6256 6332 6408 6484 6560 76 571 75- 6636 6712 6788 6864 6940 7016 7092 7168 7244 7320 227? 76 572 75- 7396 7472 7548 7624 7700 7775 7851 7927 8003 76 573 ' 75- 8155 8230 8988 8306 8382 8458 8533 8609 8685 8761 8836 76 574 , 75- 8912 9063 9139 9214 9290 9366 9441 9517 9592 76 575 75- 9668 9743 9819 9894 9970 76 575 75- • » • • 0045 0121 0196 0272 0347 75 576 75- 0422 0498 0573 0649 0724 0799 0875 0950 1025 IIOI 75 577 N 76- 1176 1251 1326 1402 1477 1552 1627 1702 1778 1853 75 D I 2 3 4 5 6 7 8 9 so MATHEMATICAL TABLES. N 578 I 2 3 4 5 6 7 8 9 D 75 76- 1928 2003 2078 2153 2228 2303 2378 2453 2529 2604 579 76- 2679 2754 2829 2904 2978 3053 • 3128 3203 3278 3353 75 580 76- 3428 3503 3578 3653 3727 3802 3877 3952 4027 4101 75 5^' 76- 4176 4251 4326 4400 4475 4550 4624 4699 4774 4848 75 5^^ 76- 4923 4998 5072 5147 5221 5296 5370 5445 5520 5594 75 553 76- 5669 5743 5818 5892 5966 6041 6115 6190 6264 6338 7082 74 584 76- 6413 6487 6562 6636 6710 6785 6859 6933 7007 74 74 585 76- 7156 7230 7304 7379 7453 7527 7601 7675 7749 7823 586 76- 7898 7972 ' 8046 8120 8194 8268 8342 8416 8490 8564 74 5^Z 76- 8638 8712 8786 8860 8934 9008 9082 9156 9230 9303 74 588 76- 9377 9451 9525 9599 9673 9746 9820 9894 9968 74 588 77- •••••• 0042 74 589 77- 0115 0189 0263 0336 0410 0484 0557 0631 0705 0778 74 590 77- 0852 0926 0999 IP73 1 146 1220 129^ 1367 1440 1514 74 591 77- 1587 1661 1734 180B 1881 ^9?S 2028 2102 2175 2248 73 592 77- 2322 2395 2468 2542 2615 2688 2762 2835 2908 2981 73 593 77- 3055 3128 3201 3274 3348 3421 3494 3567 4298 3640 3713 73 594 77- 3786 3860 3933 4006 4079 4152 4225 4371 4444 73 595 77- 4517 4590 4663 4736 4809 48S2 4955 5028 5100 5173 73 596 77- 5246 5319 5392 5465 5538 5610 5683 5756 5829 5902 73 597 77- 5974 6047 6120 6193 6265 6338 641 1 6483 6556 6629 73 598 77- 6701 6774 6846 6919 6992 7064 7137 7209 7282 7354 73 599 77- 7427 7499 7572 7644 77'7 7789 7862 7934 8006 8079 72 600 77- 8151 i^224 8296 8368 8441 8513 8585 8658 9380 8730 8802 72 601 77- 8874 8947 9019 9091 9163 9236 9308 9452 9524 72 602 77- 9596 9669 9741 9813 9885 9957 • • • • « • 72 602 78- 0029 OIOI 0173 0245 72 603 78- 0317 0389 0461 0533 0605 0677 0749 0821 0893 0965 72 604 78- 1037 IIO9 1181 1253 1324 1396 1468 1540 1612 1684 72 605 78- 1755 1827 1899 1971 2042 2114 2186 2258 2329 2401 72 606 78- 2473 2544 2616 2688 2759 2831 2902 2974 3046 3117 72 607 78- 3189 3260 3332 340.S 3475 3546 3618 3689 3761 3832 71 608 78- 3904 3975 4046 41 18 4189 4261 4332 4403 4475 4546 71 609 78- 4617 4689 4760 4831 4902 4974 5045 5116 5187 5259 71 610 78- 5330 5401 5472 5543 5615 5686 5757 5828 5899 5970 71 611 78- 6041 6112 6183 6254 6325 6396 6467 6538 6609 6680 71 612 78- 6751 6822 6893 6964 7035 7106 7177 7248 7319 7390 71 613 78- 7460 753> 7602 7673 7744 7815 7885 7956 8027 8098 71 614 78- 8168 8239 8310 8381 84s I 8522 8593 8663 8734 8804 71 615 78- 8875 8946 9016 9087 9157 9228 9299 9369 9440 9510 71 616 78- 9581 9651 9722 9792 9863 9933 70 tfi6 79- cxx}4 0074 0144 0215 70 617 79- 0285 0356 0426 0496 0567 0637 0707 0778 0848 0918 70 618 79- 0988 1059 1 129 1 199 1269 1340 1410 1480 1550 1620 70 619 620 79- I 69 I 1761 1831 1901 1971 2041 2111 2181 2252 2322 70 79- 2392 2462 2532 2602 2672 2742 2812 28S2 2952 3022 70 621 79- 3092 3162 3231 3301 3371 3441 35" 3581 3651 3721 70 622 N 79- 3790 3860 3930 4000 4070 4139 4209 4279 4349 4418 70 D I 2 3 4 _ 5 6 7 8 9 LOGARITHMS OF NUMBERS. SI Nj I « 2 3 4 5 6 7 '8 9 D 70 623 79- 4488 4558 4627 4697 4767 4836 4906 4976 5045 5"S 624 79- 5185 5254 5324 5393 5463 5532 5602 5672 5741 5811 70 625 79- 5880 5949 6019 6088 6158 6227 6297 6366 6436 6505 69 626 79- 6574 6644 6713 6782 6852 6921 6990 7060 7129 7198 69 627 79- 7268 7337 7406 7475 7545 7614 7683 7752 7821 7890 69 628 79- 7960 8029 8098 8167 8236 8305 8374 8443 8513 8582 69 629 79-8651 8720 8789 8858 8927 8996 9065 9134 9203 9272 69 630 631 79- 9341 9409 9478 9547 9616 9685 9754 9823 9892 9961 69 80- 0029 0098 0167 0236 0305 0373 0442 0511 0580 0648 69 632 80- 0717 0786 o8s4 0923 0992 1061 1 129 X198 1266 1335 69 633 80- 1404 1472 1541 1609 1678 1747 1815 1884 1952 2021 69 634 80- 2089 2158 2226 2295 2363 2432 2500 2568 2637 2705 69 635 80-2774 2842 2910 2979 3047 3116 3184 3252 3321 3389 68 636 80-3457 3525 3594 3662 3730 3798 3867 3935 4003 4071 68 637 80- 4139 4276 4344 4412 4480 4548 4616 4685 4753 68 638 8a- 4821 4889 4957 5025 5093 5161 5229 5297 5365 5433 68 639 80- 5501 5569 5637 5705 5773 5841 5908 5976 6044 6112 68 640 80- 6180 6248 6316 6384 6451 6519 6587 6655 6723 6790 68 641 80- 6858 6926 6994 7061 7129 7197 7264 7332 7400 7467 68 642 80- 7535 7603 7670 7738 7806 7873 7941 8008 8076 8143 68 643 80- 821 1 8279 8346 8414 8481 8549 8616 8684 8751 8818 67 644 80- 8886 8953 9021 9088 9156 9223 9290 9358 9425 9492 67 645 80- 9560 9627 9694 9762 9829 9896 9964 • • • • • 67 ^l 81- 0031 0098 0165 67 646 81- 0233 0300 0367 0434 0501 0569 0636 0703 0770 0837 67 ^7 81- 0904 0971 1039 1 106 "73 1240 1307 1374 144 1 1508 67 648 81- 1575 1642 1709 1776 1843 1910 1977 2044 2111 2178 67 649 1 8x- 2245 2312 2379 2445 2512 2579 2646 2713 2780 2847 67 650 81- 2913 2980 3047 3'i4 3181 i 3247 3314 3381 3448 3514 67 651 8x- 3581 3648 3714 3781 3848 ! 3914 3981 4048 4114 4181 67 652 8x- 4248 43 >4 4381 4447 45H 4581 4647 4714 4780 4847 67 J53 81^4913 4980 5046 5"3 5179 5246 5312 5378 5445 55" 66 654 655 81- 5578 5644 57" 5777 5843 , 5910 5976 6042 6109 6175 66 81- 6241 6308 6374 6440 6506 6573 6639 6705 6771 6838 66 656 81- 6904 6970 7036 7102 7169 7235 7301 7367 7433 7499 66 ^l 81- 7565 7631 7698 7764 7S30 7896 7962 8028 8094 8160 66 658 81- 8226 8292 8358 8424 8490 8556 8622 8688 8754 8820 66 659 81- 8885 8951 9017 9083 9149 9215 9281 9346 9412 9478 66 660 81- 9544 9610 9676 9741 9807 9873 9939 66 660 82- 0004 0070 0136 66 661 82- 0201 0267 0333 09S9 0399 0464 0530 0595 0661 0727 0792 66 662 82- 0S58 0924 1055 1 120 1 186 1251 1317 1382 1448 66 5^3 8a- 15 14 1579 1645 1710 1775 184E 1906 1972 2037 2103 65 664 82- 2168 2233 2299 2364 2430 2495 2560 2626 2691 2756 65 6^5 82- 2822 2887 2952 3018 3083 3148 3213 3279 3344 3409 65 666 82- 3474 3539 3605 3670 3735 3800 3865 3930 3996 4061 55 / 667 ' 83- 4126 4191 4256 4321 4386 4451 45 '6 4581 4646 47" ^J 668 82- 4776 4841 4906 4971 5036 5101 5166 5231 • 5296 5361 65 D N : I 2 3 4 5 6 7 8 9 52 MATHEMATICAL TABLES. N 669 I 2 3 4 S 6 7 8 9 D 65 82- 5426 5491 5556 5621 5686 5751 5815 5880 5945 6010 670 82- 6075 6140 6204 6269 6334 6399 6464 6528 6593 6658 65 671 82- 6723 6787 6852 6917 6981 7046 7111 717s 7240 7305 65 672 82- 7369 7434 7499 7563 7628 7692 7757 7821 7886 8531 7951 65 673 82- 8015 8080 8144 8209 8273 8918 8338 8402 8467 8595 64 674 82- 8660 8724 8789 8853 8982 9046 9111 9175 9239 64 676 82- 9304 9368 9432 9497 9561 9625 9690 9754 9818 9882 64 676 82- 9947 64 676 83- 001 1 0075 0139 0204 0268 0332 0396 0460 0525 64 677 83- 0589 obSS 0717 0781 084s 0909 0973 1037 X102 1166 64 678 83- 1230 1294 1358 1422 X486 15 JO 1614 1678 1742 1806 64 679 83- 1870 1934 1998 2062 2126 2189 2253 2317 2381 2445 64 680 83- 2509 2573 2637 2700 2764 2828 2892 2956 3020 308^ 64 681 83- 3147 3211 3275 3338 3402 3466 3530 - 3593 3657 3721 64 682 83- 3784 3848 3912 3975 4039 4103 4166 4230 4294 4357 64 683 83- 4421 4484 4548 461 1 4675 4739 4802 4866 4929 4993 64 684 83- 5056 5120 5183 5247 5310 5373 5437 5500 5564 5627 63 685 83- 5691 5754 5817 5881 5944 6007 6071 6134 6197 6261 63 686 83- 6324 6387 6451 7083 6514 6577 6641 6704 6767 6830 6894 63 687 83- 6957 7020 7146 7210 7273 7336 7399 7462 7525 63 688 83- 75»8 7652 7715 7778 7841 7904 7967 8030 8093 8156 63 689 83- 8219 8282 8345 8408 8471 8534 8597 8660 8723 8786 63 690 83- 8849 8912 8975 9038 9101 9164 9227 9289 9352 9415 63 691 83- 9478 9541 9604 9667 9729 9792 9855 9918 9981 63 691 84- • • • • 0043 63 692 84- 0106 0169 0232 0294 0357 0420 0482 0545 0608 0671 63 693 84- 0733 0796 0859 0921 0984 1046 1 109 1172 1234 1297 63 694 84- 1359 1422 1485 1547 1610 1672 1735 1797 i860 1922 63 695 84- 1985 2047 21 10 2172 2235 2297 2360 2422 2484 2547 62 696 84- 2609 2672 2734 2796 28^9 2921 2983 3046 3108 3170 62 697 84- 3233 3295 3357 3p8o 3420 3482 3544 3606 3669 3731 3793 62 698 84- 3855 3918 4042 4104 4166 4229 4291 4353 4415 62 699 84- 4477 4539 4601 4664 4726 4788 4850 4912 5532 4974 5594 5036 62 700 84- 5098 5160 5222 5284 5346 5408 5470 5656 62 701 84- 5718 S780 5842 5904 5966 6028 6090 6151 6213 6275 62 702 84- 6337 6399 6461 6523 6585 6646 6708 6770 6832 6894 62 703 84- 6955 7017 7079 7141 7202 7264 7326 7388 7449 7511 62 704 84- 7573 7634 7696 7758 7819 7881 7943 8004 8066 8128 62 705 84- 8i<89 8251 8312 8374 8435 8497 8559 8620 8682 8743 9358 62 706 84- 8805 8866 8928 8989 9051 9112 9174 9235 9297 61 707 84- 9419 9481 9542 9604 9665 9726 9788 9849 9911 9972 61 708 85- 0033 0095 0156 0217 0279 0340 0401 0462 0524 0585 61 709 85- 0646 0707 0769 0830 0891 0952 1564 1014 1075 1 136 1 197 61 710 85- 1258 1320 1381 1442 1503 1625 1686 1747 1809 61 711 85- 1870 1931 1992 2053 21 14 2175 2236 2297 2358 2419 61 712 85- 2480 2541 2602 2663 2724 2785 2846 2907 2968 3029 61 713 85- 3090 3150 3211 ^iP 3333 3394 3455 3516 3577 3637 61 714 N 85- 3698 3759 3820 3881 3941 4002 4063 4124 4185 • 4245 61 D I 2 3 ' 4 s 6 7 8 9 I' / I.OGARITHMS OF NUMBERS. 8 43^7 44^S 4488 4549 4974 5054 509s 5156 5580 5640 5701 5761 6185 624s ^306 6366 678g 68so 6910 6970 4610 4670 4731 5216 5277 5337 5822 5882 594J 6427 6487 6548 7031 7091 7152 72' siS^^ 7J93 7453 ^^3 It !p7 8597 86S7 7M I d. ^^^^ 9799 9S59 75^3 Si 16 871S 93^1 99^^ 7574 8176 8778 9379 9978 86^ 86^ 7634 8236 8838 9439 7694 7755 8297 8357 8898 8958 9499 9559 7815 7875 8417 8477 9018 9078 9619 9679 0038 0098 0158 0218 0278 86. i?^ 0996 lOSS ^^^^ '594 les^ EE mO X7'4. ^3x0 0578 1 176 1773 2370 2966 0637 0697 0757 1236 1295 1355 1833 1893 1952 2430 2489 2549 3025 3085 3144 0817 0877 H15 1475 2012 2072 2608 2668 3204 3263 7J4/S^'^^ sS^c^ A^3<^ 9525 9584 964^ 9701 9760 01 I I 0170 0228 0287 0345 0696 0755 0813 0872 0930 1281 1339 1398 1456 I5I5 1865 1923 I98I 2040 2098 S3 D 4792 4852 61 5398 5459 61 6003 6064 61 6608 6668 60 7212 7272 60 60 60 60 60 60 60 60 60 60 60 60 3620 3680 3739 3799 3858 59 4214 4274 4333 4392 4452 59 4808 4867 4926 4985 5045 59 5400 5459 5519 5578 5637 59 5992 6051 61x0 6169 6228 59 6583 6642 6701 6760 6819 59 7173 7232 7291 7350 7409 59 7762 7821 7880 7939 7998 59 8350 8409 8468 8527 8586 59 8938 8997 9056 9114 9173 59 59 59 59 58 58 2448 2506 2564 2622 2681 58 3030 3088 3146 3204 3262 58 361 1 3669 3727 3785 3844 58 4192 4250 4308 4366 4424 58 4772 4830 4888 4945 5003 58 5351 5409 5466 5524 5582 58 5929 5987 6045 6102 6160 58 6507 6564 6622 6680 6737 58 7083 7141 7199 7256 7314 58 7659 7717 7774 7832 7889 58 8234 8292 8349 8407 8464 57 8809 8866 8924 8981 9039 57 9383 9440 9497 9555 9612 57 9956 ••• 57 0013 0070 0127 0185 57 0528 0585 0642 0699 0756 57 «; 6 7 8 Q D 54 MATHEMATICAL TABLES. N 8 D 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 88- 0S14 88- 1385 88- 1955 88- 2525 88- 3093 0871 1442 2012 2581 3150 0928 1499 2069 2638 3207 0985 1556 2126 2695 3264 1042 1613 2183 2752 3321 1099 1670 2240 2809 3377 1156 1727 2297 2866 3434 1213 1784 2354 2923 3491 1271 1 841 241 1 2980 3548 1328 1898 2468 3037 3605 88- 3661 88- 4229 88- 4795 88- 5361 88- 5926 3718 4285 4852 5418 5983 3775 4342 4909 5474 6039 3832 4399 4965 5531 6096 3888 4455 5022 5587 6152 3945 4512 5078 5644 6209 4002 4569 5135 5700 6265 4059 4625 5192 5757 6321 4115 4682 5248 581 637 4172 4739 5305 5870 6434 88- 6491 88- 7054 88- 7617 88- 8179 88- 8741 6547 7111 7674 8236 8797 6604 7167 7730 8292 8853 6660 7223 7786 8348 8909 6716 7280 7842 8404 8965 6773 7336 7898 8460 9021 6829 7392 8516 9077 6885 7449 801 1 8573 9134 6942 7505 8067 8629 9190 6998 7561 8123 '8685 9246 57 57 57 57 57 57 57 57 57 56 56 56 56 56 56 775 776 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 794 805 N 88- 9302- 88- 9862 89- 89- 0421 89- 0980 89- 1537 9358 9918 0477 1035 1593 9414 9974 0533 1091 1649 9470 9526 OOJO 0589 1 147 1705 0086 0645 1203 1760 9582 9638 9694 9750 9806 0I4I 0700 1259 I8I6 0197 0756 1314 1872 0253 0812 1370 1928 0309 0868 1426 1983 0365 0924 1482 2039 89- 209s 89- 2651 89- 3207 89- 3762 89- 4316 2150 2707 3262 3817 4371 2206 2762 3318 3873 4427 2262 2818 3373 3928 4482 2317 2873 3429 3984 4538 2373 2929 3484 4039 4593 2429 2985 3540 4094 4648 2484 3040 3595 4150 4704 2540 3096 3651 4205 4759 2595 3>5i 3706 4261 4814 89- 4870 89- 5423 89- 5975 89- 6526 89- 7077 4925 6030 6581 7132 4980 5533 6085 6636 7187 5036 5588 6140 6692 7242 5091 5644 6195 6747 7297 5146 5699 6251 6802 7352 5201 5754 6306 6857 7407 5257 5809 6361 6912 7462 5312 5864 6416 6967 7517 5367 5920 6471 7022 7572 8g- 7627 89- 8176 89- 8725 89- 9273 89- 9821 90- 7682 8231 8780 9328 9875 7737 82S6 8835 9383 9930 7792 8341 8890 9437 9985 7847 8396 8944 9492 0039 795 90- 0367 0422 0476 0531 0586 796 90- 0913 0968 1022 1077 1 131 797 90- 1458 1513 1567 1622 1676 798 90- 2003 2057 21 12 2166 2221 799 90- 2547 2601 2655 2710 2764 7902 8451 8999 9547 7957 8506 9054 9602 8012 8561 9109 9656 8067 8615 9164 9711 8122 8670 9218 9766 0094 0149 0203 0258 0312 0640 1186 1731 2275 2818 0695 1240 1785 2329 2873 0749 1295 1840 2384 2927 0804 1349 1894 2438 2981 0859 1404 1948 2492 3036 800 90- 3090 3144 3199 3253 3307 801 90- 3633 3687 3741 3795 3849 802 90- 4174 4229 4283 4337 4391 803 90- 4716 4770 4824 4878 4932 804 90- 5256 5310 5364 5418 5472 90- 5796 5850 5904 5958 6012 3361 39<H 4445 4986 5526 3416 3958 4499 5040 5580 3470 4012 4553 5094 5634 3524 4066 4607 5148 5688 3578 4120 4661 5202 5742 56 56 56 56 56 56 56 55 55 55 55 55 55 ii 55 55 55 55 55 55 55 55 54 54 54 54 54 54 54 54 6066 6119 6173 6227 6281 54 ^ 6 7 S I) LOGARITHMS OF NUMBERS. 55 N 8 D 'So6 ■807 808 ;809 90- 6335 6389 90- 6874 6927 90- 7411 7465 90- 7949 8002 6443 6497 6551 6981 7035 7089 75*9 7573 7626 S056 81 10 8163 6604 7143 7680 8217 6658 7196 7734 8270 6712 7250 7787 8324 6766 7304 7841 8378 6820 7358 7«95 8431 810 811 812 8f2 814 90- 8485 8539 9&- 9021 9074. 90- 9556 9610 y*^ «>•••• m m 9 m m m 91- 0091 0144. 91- 0624 0678 8592 8646 9128 9181 9663 9716 0197 0251 0731 0784 8699 9235 9770 0304 0838 8753 9289 9823 0358 0891 8807 9342 9877 041 1 0944 8860 9396 9930 0464 0998 8914 9449 9984 0518 105 1 8967 9503 0037 0571 1104 54 54 54 54 54 54 54 53 53 53 815 816 817 :8ig 819 91- 1158 121 1 1264. 91-1690 1743 '797 91- 2222 2275 ^3^^ 91- 2753 2806 2»59 9»- 3284 3337 3390 1317 1850 2381 29<3 3443 1371 1903 2435 2966 3496 1424 1956 2488 3019 3549 1477 2009 2541 3072 3602 1530 2063 2594 3125 3655 1584 2116 2647 3178 3708 1637 2169 2700 3231 3761 820 821 822 823 824 91- 91- 91- 91- 91- 3814 4343 4872 S40O 5927 3867 439^ 4925 5453 5980 3920 4449 4977 6033 3973 4502 5030 5558 6085 4026 4555 5083 561 1 6138 4079 4608 5136 5664 6191 4132 4660 5189 5716 6243 4184 4713 5241 5769 6296 4237 4766 5294 5822 6349 4290 4819 5347 5875 6401 825 826 827 i 828 829 91- 91- 9X- 91- 91- 6454 6980 7506 8030 8555 6507 7033 7558 8083 8607 6559 7085 761 1 8x35 8659 6612 7138 7663 8188 8712 6664 7190 7716 8240 8764 6717 7243 7768 8293 8816 6770 7295 7820 8345 8869 6822 7348 7873 8397 8921 6S75 7400 7925 8450 8973 6927 7453 7978 8502 9026 830 ! 9X-9078 9130 831 91- 9601 9653 831 92- 832 92- 0123 0176 833 92- 0645 0697 834 9a- 1166 1218 9183 9706 0228 0749 1270 9235 9758 0280 0801 1322 9287 9810 0332 0853 1374 9340 9862 0384 0906 1426 9392 9914 0436 0958 1478 9444 9967 0489 lOIO 1530 9496 9549 0019 0541 1062 1582 0071 0593 1 1 14 1634 835 836 838 839 92- 1686 92- 2206 9a- 2725 92- 3244. 92- 3762 1738 2258 ^777 3296 3814 1790 2^10 2829 m 1842 2362 2881 3399 3917 1894 2414 2933 3451 3969 1946 2466 2985 3503 4021 1998 2518 3037 3555 4072 2050 2570 3089 3607 4124 2102 2622 3140 3658 4176 2154 2674 3192 3710 4228 840 841 842 ,843 ' 844 9»- 4279 92- 4796 92- 5312 92- 5828 92- 6342 4848 53^4 5879 6394 4383 4899 5415 593 « 6445 4434 4951 5467 5982 6497 4486 5003 5518 6034 6548 4538 5054 5570 6p85 6600 4589 5106 5621 6137 6651 4641 5157 5673 6188 6702 4693 5209 5725 6240 6754 4744 5261 5776 6291 6805 845 1846 I 847 '848 I 849 92- 6857 92- 7370 92- 78S3 92- 8396 92- 8908 6908 7422 7935 8447 8959 6959 7473 7986 8498 9010 701 1 7524 8037 8549 9061 7062 7576 8088 8601 9112 7114 7627 8140 8652 9163 7165 7678 8191 8703 9215 7216 7730 8242 8754 9266 7268 7781 8293 8805 9317 73^9 7832 8345 8857 9368 93- 9419 92- 9930 9470 9981 9521 9572 9623 9674 9725 9776 9827 9879 7 8 53 53 53 53 53 53 53 53 53 53 53 53 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 51 51 51 51 51 51 51 51 51 D S6 MATHEMATICAL TABLES. N I 2 3 4 5 6 7 8 9 D 851 93- 0032 0083 0134 0185 0236 0287 0338 0389 0S47 0898 51 852 93- 0440 0491 0542 0592 0643 0694 0745 0796 51 853 93- 0949 IOCX> 105 1 1 102 1153 1203 1254 1305 1356 1407 1865 1915 SI 854 93- 1458 1509 1560 1610 1661 1712 1763 1814 51 51 856 93- 1966 2017 2068 2118 2169 2220 2271 2322 2372 2423 856 93- 2474 2524 2575 2626 2677 2727 2778 2829 2879 2930 SI 857 93- 2981 3031 3082 3133 3183 3234 3285 3335 3386 3437 3892 3943 4397 4448 SI 858 93- 3487 3538 3589 3639 3690 3740 3791 3841 51 859 93- 3993 4044 4094 4145 4195 4246 4296 4347 SI 860 93- 4498 4549 4599 4650 4700 4751 4801 4852 4902 4953 50 861 93- 5003 5054 5104 5154 5205 5255 5306 5809 5356 5406 5457 50 862 93- 5507 5558 5608 5658 5709 5759 5860 5910 5960 50 863 93- 601 I 6061 6111 6162 6212 6262 6313 6815 6363 6865 6413 6463 50 864 93- 6514 6564 6614 6665 6715 6765 6916 6966 50 865 93- 7016 7066 7117 7618 7167 7217 7267 7317 7367 7418 7468 50 866 93- 75>8 7568 7668 7718 7769 8269 7819 7869 7919 7969 50 867 93- 8019 8069 8119 8169 8219 8319 8370 8420 8470 50 868 93- 8520 8570 8620 8670 8720 8770 8820 8870 8920 8970 so 869 93- 9020 9070 9120 9170 9220 9270 9320 9369 9419 9469 50 870 93- 9519 9569 9619 9669 9719 9769 9819 9869 9918 9968 50 871 94- 0018 0068 0118 0168 0218 0267 0317 0815 0367 0865 0417 0467 50 872 94- 0516 0566 0616 0666 0716 0765 0915 0964 50 873 94- 1014 1064 1 1 14 1 163 1213 1263 1809 1362 1412 1462 50 874 94- 1511 1561 1611 1660 1710 1760 1859 1909 1958 50 875 94- 2008 2058 2107 2157 2207 2256 2306 2801 2355 2405 2455 50 876 94- 2504 2554 2603 2653 2702 2752 2851 2901 2950 50 ^77 878 94- 3000 3049 3099 3148 3198 3247 3297 3346 3396 3445 49 94- 3495 3544 408!; 3^43 3692 3742 3791 3841 3890 3939 49 879 94- 3989 4038 4137 4186 4236 4285 4335 4384 4433 49 880 94- 4483 4532 4581 4631 4680 4729 4779 4828 4877 4927 49 881 94- 4976 5025 5074 5124 5>73 5222 5272 5321 5370 5419 5862 59 r2 49 882 94- 5469 5518 5567 5616 5665 5715 5764 5813 49 883 94- 5961 6010 6059 6108 6157 6207 6256 6305 6354 6403 6845 6894 49 884 94- 6452 6501 6551 6600 6649 6698 6747 6796 49 49 885 94- 6943 6992 7041 7090 7140 7189 7238 7287 7336 7385 886 94- 7434 7483 7532 7581 7630 7679 7728 7777 7826 7875 49 887 94- 7924 7973 8022 8070 8119 8168 8217 8266 8315 8364 8804 8853 49 888 94- 8413 8462 8511 8560 8609 8657 8706 8755 49 889 94- 8902 8951 8999 9048 9097 9146 9195 9244 9292 9341 49 890 94- 9390 9439 9488 9536 9585 9634 9683 9731 9780 9829 49 891 94- 9878 9926 9975 49 891 95- 0024 0073 OI2X 0170 0219 0267 0316 0754 0803 49 892 95- 0365 95- 0851 0414 0462 0511 0560 0608 0657 0706 49 893 0900 0949 0997 1046 1095 "43 1 192 1240 1289 49 894 95- 1338 1386 1435 1483 1532 1580 1629 1677 1726 1775 49 896 95- 182J ^ 95- 2308 1872 1920 1969 2017 2066 21 14 2163 22 I I 2260 48 896 2356 ^?5 2453 2502 2550 2599 2647 2696 2744 48 897 N 95- 2792 2841 2889 2938 2986 3034 3083 3131 3180 3228 48 I 2 3 4 5 6 7 8 9 D LOGARITHMS OF NUMBERS. 57 i N 8 D 48 48 48 48 48 48 48 898 95- 3276 3325 3373 3421 3470 899 ! 95- 3760 3808 3856 3905 3953 3518 4001 3566 4049 361. 409^ 3663 4146 37" 4194 900 ' 95- 4243 4291 4339 901 I 95- 4725 4773 4821 95- 5207 5255 5303 95- 5688 5736 5784 95- 6168 6216 6265 902 903 904 905 906 ,907 I 908 909 4387 4869 5351 32 313 4435 4918 5399 5880 6361 95- 6649 95- 7128 95- 7607 95- 8086 95- 8564 6697 7176 7655 8134 8612 6745 7224 7703 8181 8659 6793 7272 7751 8229 8707 6840 7320 7799 8277 8755 4484 4966 5447 5928 6409 4532 5014 5495 5976 6457 4580 5062 5543 6024 6505 4628 5110 5592 6072 6553 4677 5158 5640 6120 6601 6888 7368 7847 8325 8803 6936 7416 7894 8373 8850 6984 7464 7942 8421 88q8 7032 7512 7990 8468 8946 7080 7559 8038 8516 8994 48 48 48 48 48 48 48 48 48 48 47 910 911 ! 912 912 913 914 95- 9041 95- 9518 95-9995 96- 96- 0471 96- 0946 9089 9566 0042 0518 0994 9137 9614 0090 0566 1041 9185 9661 0138 0613 1089 9232 9709 0185 066 X 1136 9280 9757 0233 0709 1184 9328 9804 • • • « 0280 0756 1231 9375 9852 0328 0804 1279 9423 9900 0376 0851 1326 9471 9947 Q423 0899 1374 916 I 96- 1421 916 917 918 919 96- 1895 96- 2369 96- 2843 9&- 33^6 1469 1943 2417 2890 3363 1516 1990 2464 2937 3410 1563 2038 251 1 2985 3457 1611 2085 2559 3032 3504 1658 2132 2606 3079 3552 1706 2180 2653 3126 3599 1753 2227 2701 3174 3646 1801 2275 2748 3221 3693 1848 2322 2795 3268 3741 47 47 47 47 47 920 921 922 923 924 96-3788 96- 4260 96-4731 96- 5202 96- 5672 3835 4307 4778 5249 5719 3882 4354 4825 5296 5766 3929 4401 4872 5343 5813 3977 4448 4919 5390 5^ 4024 4495 4966 5437 5907 4071 4542 5013 5484 5954 4118 4590 5061 5531 6001 4165 4637 5108 6048 4212 i 47 4684 5155 5625 6095 47 47 47 47 925 926 927 928 929 930 1931 ' 932 933 933 .934 i" 985 936 937 938 939 940 941 942 943 IT 96- 6142 96- 661 I 96- 7080 96- 754S 96- 8016 6189 6658 7127 7595 8062 6236 6705 7173 7642 8109 6283 6752 7220 7688 8156 6329 6799 7267 7735 8203 6376 6845 7314 7782 8249 6423 6892 7361 7829 8296 6470 6939 7408 7875 8343 6517 6986 7454 7922 8390 6564 7033 7501 7969 8436 96-8483 96- 8950 96- 9416 96- 9882 97- 97- 0347 8530 8996 9463 9928 8576 9043 9509 9975 0393 0440 8623 9090 9556 0021 0486 8670 9136 9602 0068 0533 8716 9183 9649 01 14 0579 8763 9229 9695 0161 0626 8810 9276 9742 0207 0672 8856 9323 9789 0254 0719 8903 9369 9835 0300 0765 97- 0812 97- 1276 97- 1740 97- 2203 97- 2666 0858 1322 1786 2249 2712 0904 1369 1832 2295 2758 0951 1415 1879 2342 2804 0997 1461 1925 2388 2851 1044 1508 1971 2434 2897 1090 >554 2018 2481 2943 "37 1601 2064 2527 2989 1x83 1647 21 10 2573 3035 97- 3128 97- 3590 97- 4051 97- 4512 3174 3636 4097 4558 3220 3682 4143 4604 3266 3728 4189 4650 3313 3774 4235 4696 3359 3820 4281 4742 3405 3866 4327 4788 3451 3913 4374 4834 3497 3959 4420 4880 8 1229 1693 2157 2619 3082 3543 4005 4466 4926 47 47 47 47 47 47 47 47 47 47 46 46 46 46 46 46 46 46 46 ii D - "-It. '-/*f /■ • 58 MATHEMATICAL TABLES. N 944 I 2 3 4 5 6 7 8 9 D 46 97- 4972 5018 5064 5110 5156 5202 5248 5294 5340 5386 945 97- 5432 5478 5524 5570 5616 5662 5707 5753 • 5799 5845 46 946 97- 5891 5937 5983 6029 6075 6121 6167 6212 6258 6304 46 947 97- 6350 97- 6808 6396 6442 64S8 6533 6579 6625 6671 6717 6763 46 948 6854 6900 6946 6992 7037 7083 7129 7175 7220 46 949 97- 7266 7312 7358 7403 7449 7495 7541 7586 7632 7678 46 960 97- 7724 7769 7815 7861 7906 7952 8409 7998 8454 8043 8089 8135 46 951 97- 8i8i 8226 8272 8317 8363 8500 8546 8591 46 952 97- 8637 8683 8728 8774 8819 8865 891 1 8956 9002 9047 46 953 97-9093 9138 9184 9230 9275 9321 9366 9412 9457 9503 46 954 97- 9548 9594 9639 9685 9730 9776 982 X 9867 9912 9958 46 966 98- 0003 0049 0094 0140 D185 0231 0276 0322 0367 082 X ^l^ 45 956 98- 0458 0503 0549 0594 0640 0685 0730 0776 0867 45 957 98- 0912 0957 1003 1048 1093 "39 1x84 1229 1275 1320 45 958 98- 1366 X411 1456 1 501 1547 1592 1637 1683 1728 1773 45 959 98- 1819 1864 1909 1954 2000 2Q45 2090 2135 2181 2226 45 960 98- 2271 2316 2362 2407 2452 2497 2543 2588 2653 3085 2678 45 961 98- 2723 2769 2814 2859 2904 2949 2994 3040 3130 45 962 98- 3175 3220 3265 33 JO 3356 3807 3401 3446 3491 3536 3581 45 963 98- 3626 3671 3716 3762 3852 3897 3942 3987 40J2 45 964 98- 4077 4122 4167 4212 4257 4302 4347 4392 4437 4482 45 965 98- 4527 4572 4617 4662 4707 4752 4797 4842 4887 4932 45 966 98- 4977 5022 5067 5112 5157 5202 5247 5292 5337 5382 45 967 98- 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 45 968 98- 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 45 969 98- 6324 6369 6413 6458 6503 6548 6593 6637 6682 6727 45 970 98- 6772 6817 6S61 6906 6951 6996 7040 7085 7130 7175 45 971 98- 7219 7264 7309 7353 7800 7398 7443 7488 7532 7577 7622 45 972 98- 7666 77" 7756 7845 7890 7934 7979 8024 8068 45 973 98- 81 13 8157 8202 8247 8291 8336 838X 8425 8470 8514 45 974 98- 8559 8604 8648 8693 8737 8782 8826 8871 8916 8960 45 975 98- 9005 9049 9094 9138 9185 9227 9272 9316 9361 9405 45 976 98- 9450 9494 9539 9583 9628 9672 9717 9761 9806 9850 44 977 98- 9895 9939 9983 • • • 44 977 99- 0028 0072 0117 0161 0206 0250 0294 44 978 99- 0339 0383 0428 0472 0516 0561 0605 0650 0694 0738 44 979 99- 0783 0827 0871 0916 0960 1004 1049 1093 1137 1 182 44 980 99- 1226 1270 I3J5 1359 1403 1448 1492 1536 1580 1625 44 981 99- 1669 1713 1758 1802 1846 ! 1890 1935 1979 2023 2067 44 982 99- 21 I I 2156 2200 2244 2288 2333 2377 2421 2465 2509 44 983 99- 2554 99- 2^95 2598 2642 2686 2730 2774 2819 2863 2907 2951 44 984 3039 3083 3127 3172 3216 3260 3304 3348 3392 44 985 99- 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 1 44 986 99- 3877 3921 3965 4009 4053 4097 4141 4185 4229 4273 44 987 99- 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 44 988 99- 4757 4801 4845 4889 4933 4977 5021 5065 5108 5152 44 989 N 99- 5196 5240 5284 5328 5372 5416 5460 5504 5547 5591 44 D I 2 3 4 5 6 7 8 9 LOGARITHMS OF NUMBERS. 59 N I 2 3 4 5 6 7 8 9 D 990 9^- 5635 5679 5723 5767 5811 5854 5898 5942 5986 6030 44 991 9^- 6074. 6117 6161 6205 6249 6293 6337 6380 6424 6468 44 992 99- 6512 6555 6599 6643 6687 6731 6774 6818 6862 6906 44 993 9Q- 694.9 6993 7037 7080 7124 7168 7212 7255 7299 7343 44 994 99- 73S^ 7430 7867 7474 7910 7517 7954 7561 7998 7605 8041 7648 7692 7736 7779 44 44 995 99- 7823 8085 8129 8172 8216 996 99- 8259 8303 8347 83QO 8434 8477 8521 8564 8608 8652 9087 44 997 99- 8695 8739 8782 8826 8869 8913 8956 9000 9043 44 99» 99- 9131 9174 9218 9261 9305 93*8 9392 9435 9479 9522 44 999 99- 9565 9609 9652 9696 9739 9783 9826 9870 9913 9957 43 D N I 2 3 4 5 6 7 8 9 / 6o MATHEMATICAL TABLES. TABLE No. IL— HYPERBOLIC LOGARITHMS OF NUMBERS FROM 1. 01 TO 30. Number. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. I.OI 1.02 1.03 1.04 1.05 .0099 .0198 .0296 .0392 .0488 1.36 1-37 1.38 1-39 1.40 •3075 .3148 .3221 .3293 •3365 I.7I 1.72 1-73 1.74 1 1.75 •5365 .5423 .5481 •5539 •5596 2.06 2.07 2.08 2.09 2.10 .7227 •7275 •7324 •7372 .7419 1.06 1.07 1.08 1.09 1. 10 .0583 .0677 .0770 .0862 ■0953 1.41* 1.42 1-43 1.44 1.45 .3436 .3507 .3577 .3646 •3716 1.76 1.77 1.78 1.79 1.80 .5653 .5710 .5766 .5822 .5878 2. II 2.12 2.13 2.14 2^15 .7467 •7514 .7561 .7608 .7655 1. 11 1. 12 ^•13 1. 14 ^•15 .1044 .1133 .1222 .1310 .1398 1.46 1.47 1.48 1.49 1.50 .3784 .3853 .3920 .3988 •4055 1.81 1.82 1.83 1.84 1.85 .5933 .5988 .6043 .6098 .6152 2.16 2.17 2.18 2.19 2.20 .7701 .7747 •7793 .7839 .7885 1. 16 1.17 1. 18 1. 19 1.20 .1484 .1570 .1655 .1740 .1823 I-5I 1-52 1-53 1^54 1.55 .4121 .4187 •4253 .4318 ■4383 1.86 1.87 1.88 1.89 1.90 .6206 .6259 .6313 .6366 .6419 2.21 2.22 2.23 2.24 2.25 •7930 .7975 .8020 .8065 .8109 1. 21 1.22 1.23 1.24 1.25 .1906 .1988 .2070 .2151 .2231 1.56 1-57 1.58 1.59 1.60 .4447 .4511 •4574 •4637 .4700 1.91 1.92 1.93 1.94 1.95 .6471 •6523 •6575 .6627 .6678 2.26 2.27 2.28 2.29 2.30 .8154 .8198 .8242 .8286 •8329 1.26 1.27 1.28 1.29 1.30 .2311 .2390 .2469 .2546 .2624 1.61 1.62 1.63 1.64 1.65 .4762 .4824 .4886 .4947 .5008 1.96 1.97 1.98 1.99 2.00 .6729 .6780 .6831 .6881 .6931 2.31 2.32 2.33 2.34 2.35 .8372 .8416 .8458 .8502 .8544 131 1.32 1.33 1.34 I.3S .2700 .2776 .2852 .2927 .3001 1.66 1.67 1.68 1.69 1.70 .5068 .5128 .5188 .5247 .5306 2.01 2.02 2.03 2.04 2.05 .6981 •7031 .7080 .7129 .7178 2.36 2.37 2.38 2.39 2.40 .8587 .8629 .8671 •8713 .8755 j HYPERBOLIC LOGARITHMS OF NUMBERS 6l 1 4 !f umber. Logarithm. Number. Logarithm. Number. Logarithm. 1 Number. Logarithm. 2.41 .8796 2.8l 1.0332 3.21 I.1663 3.61 1.2837 2.42 .8838 2.82 1.0367 3.22 1. 1694 3.62 1.2865 2.43 .8879 2.83 1.0403 3.23 I.I725 3.63 1.2892 2.44 .8920 2.84 1.0438 3.24 I.1756 3-64 1.2920 2.45 .8961 2.85 1.0473 3.25 I.I787 3.65 1.2947 2.46 .9002 2.86 1.0508 3.26 1.1817 3.66 1.2975 2.47 .9042 2.87 1-0543 3-27 1. 1848 3.67 1.3002 2.48 1 -9083 2.88 1.0573 328 1. 1878 3.68 1.3029 2.49 1 .9123 2.89 I.0613 3.29 1. 1 909 3.69 1.3056 2.50 -9163 2.90 1.0647 3.30 1-1939 370 1.3083 2.51 .9203 2.91 1.0682 3.31 1. 1969 3.71 1.3110 2-5* -9243 2.92 1.0716 3.32 1. 1999 372 I.3137 *-53 .92^2 2.93 1.0750 3-33 1.2030 3.73 I.3164 2.54 .9322 2.94 1.0784 3-34 1.2060 3.74 I.319I 2-55 .9361 2.95 i.o8i8 3.35 1.2090 3.75 1.3218 2.56 .9400 2.96 1.0852 3.36 1.2119 3.76 1.3244 2.57 •9439 2.97 1.0886 3-37 1. 2149 3.77 I.3271 2.58 .9478 2.98 1.0919 3.38 I.2179 3.78 1.3297 1 2.59 .9517 2.99 1.0953 3-39 1.2208 3.79 1.3324 2.60 -9555 3.00 1.0986 3.40 1.2238 3.80 1-3350 2.61 -9594 3.01 1.1019 3.41 1.2267 3.81 1.3376 2.62 .9632 3.02 1. 1053 3.42 1.2296 3.82 1.3403 2.63 .9670 303 1. 1 086 3-43 1.2326 383 1.3429 2.64 .9708 304 1.1119 3-44 1.2355 3.84 1-3455 2.65 .9746 3-05 1.1151 3.45 1.2384 3.85 1.3481 2.66 .9783 3.06 1.1184 3.46 1.2413 3.86 1-3507 2.67 .9821 3-07 1.1217 3.47 1.2442 3.87 1-3533 2.68 .9858 3.08 1. 1249 3.48 1.2470 3.88 1.3558 1 2-69 .9895 .309 1.1282 3-49 1.2499 3.89 1-3584 2.70 -9933 3.10 1.1314 350 1.2528 3.90 1.3610 2.71 .9969 3" 1. 1346 3.51 1.2556 3-91 1.3635 2.72 1.0006 3.12 1.1378 352 1.2585 3.92 1.3661 2.73 1.0043 313 1.1410 3.53 1. 2613 3.93 1.3686 2.74 1.0080 3.14 1.1442 3.54 1. 2641 3.94 1.3712 2.75 1.0116 315 1.1474 3.55 1.2669 3.95 . 1.3737 ^ 2.76 1-0152 3.16 1. 1506 3.56 1.2698 3.96 1.3762 2.77 1.0188 3-17 1.1537 3.57 1.2726 3-97 1.3788 2.78 1.0225 3.18 1.1569 3.58 1.2754 3.98 1.3813 2.79 1.0260 319 1. 1600 3-59 1.2782 3-99 1.3838 2.80 1.0296 3.20 1.1632 3.60 1.2809 4.00 1.3863 ! MATHEMATICAL TABLES. Logiiithm. I Numbe. Logarithm. Numbn. L.«lfithlB. Numbn. l.:g..ilhn.- 1.3888 4.41 1.4839 4.8. 1-5707 S-2I .-6506 '■39'3 1 4-42 ..4861 4.82 1-5728 S-22 1-6525 '■3938 i 4-43 r.4884 4.83 1-5748 5-23 .-65.4 1.3962 ; 4-44 1.4907 4.84 1.5769 S-24 .6563 1-3987 ,4.45 1.4929 4.85 1.5790 5-25 .-6582 1. 401 2 , 4.46 1-495' 4.86 1.5S.0 5-26 .-6601 1.4036 ! 4-47 1.4974 4.87 ..583. 5-27 1-6620 ..4o6r 4.48 1.4996 4.88 1-5851 5.28 1-6639 1.408s 4.49 I.50I9 4.89 ..5872 5.29 1-6658 1.4110 1 4.50 1.5041 4.90 1.5892 5.30 1-6677 1.4134 ! 4-51 1.5063 4.91 1-5913 5-31 1-6696 i-4'S9 ; 4.52 1.5085 4.92 1-5933 S-32 I-67I5 1.4183 i 453 1.5107 4.93 1-5953 5-33 1-6734 1.4207 4-54 1.5129 4-94 1-5974 S-34 1-6752 1.4231 4.55 1.5151 4.95 1-5994 5-35 1-6771 I-42SS i 4.56 i.S"73 4.96 1-6014 5-36 1-6790 1.4279 , 4.57 '■S'95 4.97 1-6034 S-37 1-6808 1.4303 ,4.58 1-5217 4.98 1-6054 5-38 1-6827 1-4327 1 4-59 1.5239 499 1-6074 5-39 1-6845 I-43S1 I4.60 1-526. 5.00 1-6094 5-40 I-6S64 •■4375 , 4.61 1.5282 ! 5-01 1-6.14 S-41 1-6882 1.4398 4.62 1-5304 5.02 1-6134 5-42 1-690, 1.4422 4.63 1.5326 5-03 1-6154 5-43 I-69I9 1.4446 4-64 1.5347 5.04 .-6,74 5-44 1-6938 1.4469 , 4-65 1.5369 S-OS .-6194 S-45 1-6956 1-4493 4.66 ■■5390 5.06 1-6214 5-46 1-6974 1.4516 4.67 1.5412 5.07 .6233 5-47 1-6993 1.4540 4.68 1.S433 5.08 .-6253 S-48 1-7011 1.4563 4.69 1.5454 5.09 1-6273 5-49 1-7029 1.4586 4.70 1.5476 5.1" 1-6292 5-50 1-7047 1.4609 4.71 1.5497 511 .-6312 5-51 1-7066 1-4633 4-72 1.5518 5.12 .-6332 5-52 1-7084 1.4656 4-73 1 5539 513 1-635. S-53 I-7I02 1.4679 4-74 1.5560 5.14 1637. 5-54 1-7.20 1.4702 4-75 1.55s. 5.15 1-6390 5-55 .-7.38 1-4725 4.76 1.5602 5.16 1-6409 5-s6 1-7156 '.4748 4.77 1.5623 517 ,6429 5-57 1-7174 1.4770 4.78 1.5644 5.18 1-6448 5.5s 1-7192 1-4793 4-79 1.5665 5.19 ,-6467 5-59 1-7210 1.48.6 4.80 1.5686 5.20 .-6487 5-60 1-7228 HYPERBOLIC LOGARITHMS OF NUMBERS. 63 i Number. Logarithm. Number. 6.01 6.02 6.03 6.04 6.05 Logarithm. Number. Logarithm. Number. Logarithm. 5.61 1.7246 5.62 1.7263 5.63 1. 7281 •5.64 1.7299 5-65 I 1. 7317 1 1.7934 I-795I 1.7967 1.7984 1. 800 1 6.41 6.42 6.43 6.44 6.45 1.8579 1.8594 I.8610 1.8625 1. 8641 6.81 6.82 6.83 6.84 6.85 1.9184 I.9199 I.9213 1.9228 1.9242 5-66 5.67 5.68 5-69 5-70 1-7334 1-7352 1.7370 1-7387 1.7405 6.06 6.07 6.08 6.09 6.10 1. 8017 1.8034 1.8050 1.8066 1.8083 i 6.46 6.47 6.48 6.49 6.50 1.8656 1.8672 1.8687 1.8703 I.8718 6.86 6.87 6.88 6.89 6.90 1.9257 1.9272 1.9286 I.93OI 1.9315 5-71 5-72 5-73 5-74 1 5-75 1 1.7422 1.7440 1.7457 1.7475 1.7492 6.U 6.12 6.13 6.14 6.15 1 1.8099 1.8116 1.8132 1.8148 1.8165 6.51 6.52 6.53 ' 6.54 : 6.55 1.8733 1.8749 1.8764 1.8779 1.8795 6.91 6.92 6.93 i 6.94 6.95 1.9330 1.9344 1.9359 1-9373 1-9387 5-76 5-77 5-78 5-79 5-80 1.7509 1.7527 1.7544 I. 7561 1.7579 6.16 6.17 6.18 6.19 6.20 i.8i8i 1.8197 1.8213 1.8229 1.8245 6.56 6.57 6.58 6.59 • 6.60 1 1.8810 1.8825 1.8840 1.8856 1.8871 6.96 ! 6.97 6.98 6.99 7.00 1.9402 1. 9416 L 1.9430 1.9445 1.9459 5-81 5-8* 5-83 584 5-85 1.7596 I. 7613 1.7630 1.7647 1 . 7664 6.21 6.22 6.23 6.24 6.25 1.8262 1.8278 1.8294 1.8310 1.8326 6.61 6.62 6.63 6.64 6.65 1.8886 1. 8901 1. 8916 1.8931 1.8946 7.01 7.02 7.03 7.04 7.05 1-9473 1.9488 1.9502 1.9516 1.9530 5-86 5-87 5.88 5-89 5.90 1. 7681 1.7699 1. 7716 1-7733 1.7750 6.26 6.27 6.28 6.29 6.30 1 1.8342 1.8358 1-8374 1.8390 1.8405 6.66 6.67 6.68 6.69 6.70 1. 8961 1.8976 1.8991 1.9006 1. 9021 7.06 7.07 7.08 7.09 7.10 1.9544 1.9559 1.9573 1.9587 1. 9601 5-91 5-92 5-93 5-94 5-95 1.7766 1.7783 1.7800 1. 7817 1.7834 1 1 6.31 6.32 6.33 6.34 6.35 1. 8421 1.8437 1.8453 1.8469 1.8485 ' 6.71 6.72 6.73 6.74 6.75 1.9036 1.9051 1.9066 1. 9081 1.9095 7.II 7.12 7.13 7.14 7.15 1.9615 1.9629 1.9643 1.9657 1.9671 5-96 5-97 5.98 5-99 . 6.00 1 • 1.7851 1.7867 1.7884 1.7901 1. 7918 6.36 6.37 6.38 6.39 6.40 1.8500 1.8516 1.8532 1.8547 1.8563 6.76 6.77 . 6.78 1 6.79 6.80 1.9110 1.9125 1. 9140 1.9155 1. 9169 7.16 7.17 7.18 7.19 7.20 1.9685 1.9699 1.9713 1.9727 1. 9741 MATHEMATICAL TABLES. I^,g«i*m. Number. ^.^. Number. Logarilhn.. »„,»,. L.«uithin. I-97SS 1.9769 ..9782 7,61 7.62 7-63 7-64 7-65 2.0295 2.0308 2.0321 Z0334 2-0347 8.01 8.02 8.03 8.04 8.05 2.0807 2.0819 2.0832 2.0844 2.0857 8.41 8.42 8.43 8.44 8-45 2.1294 2.1330 2.1342 1.9824 1.9838 1.9851 1.986s 1.9879 7.66 7.67 7.68 7.69 7.70 2.0360 2-0373 2.0386 2.0399 2.0412 8.06 8.07 8.08 8.09 8.10 2.0869 2.0882 2.0894 2.0906 2.0919 S.46 8.47 8.48 If. 21353 2.1365 2.1377 2.1389 2.1401 ■.9892 1.9906 1.9920 1.9933 1-9947 7.71 7.72 7-73 7.74 7-75 2-0425 2-0438 2.0451 2.0464 2.0477 8.1 1 8.12 8.13 8.14 8.15 2.0931 2.0943 2.0980 8.51 8.52 8.53 8.54 8.SS 2.1412 2.1424 2.1436 2.1448 2.1459 ,.996, ■9974 ..9988 2.0001 2.0015 7.76 7-77 7-78 7-79 7.80 2.0490 2.0503 2.0516 2.0528 2.0541 8.16 8.17 8.18 8.19 8.30 2.0992 2.I00S 2.1017 2.1029 2.IO4I 8.56 8.57 8.58 8.59 8.60 2.1471 2.1483 2.1494 2.1506 2.1518 2.0028 2.0042 2-005 S 2.0069 2.0082 7.81 7.82 7-83 7-84 7-8s 2.0554 2.0567 2.0580 2.0605 8.21 8.22 8.23 8.24 8.25 2.1054 2,1066 2.1078 2. 1090 8.61 8.62 8.63 8.64 8.65 2.1529 2.1541 2.1552 2.1564 2.15,6 2.0096 2.0109 2.0149 7.86 7-87 7.88 7.89 7.90 2.0618 2.0631 2.0643 2.0656 2.0669 8.26 8.27' 8.28 8.29 8.30 2!lI26 2.1138 2. I 150 2.I163 8.66 8.67 8.68 8.69 8.70 2.1587' 2.1599 2.1610 2.1622 2.1633 2.0162 2.0176 2'0202 2.0215 7.91 7.92 7-93 7-94 7-95 2.0681 2.0694 2.0707 2.0719 2.0732 8.31 8.32 8.33 8.34 8-3S 2.H99 2.12II 2.1223 8.71 8.72 8.73 8.74 8.75 2.1645 2.1656 2.1668 2.1679 2.169, 2.0229 2.0242 2-02SS 2.0268 2.0281 7.96 7.97 7.98 7-99 8.00 2.0744 2.0757 2.0769 2.0782 2-0794 8.36 8.37 8.38 8.39 8.40 2.1235 2.1247 2.1258 2.1270 2.1282 8.,6 8-77 8.78 8.79 8.80 2.1702 2.1713 2.1725 2.1,36 2.1748 HYPERBOLIC LOGARITHMS OF NUMBERS. 65 t Nanber. Logarithm. Number. Logarithm. Number. Logarithm. Number. Logarithm. 8.81 2.1759 9.II 2.2094 9.41 2.2418 9.71 2.2732 8.82 2.1770 9.12 2.2105 9.42 2.2428 9.72 2.2742 8.83 2.1782 9-13 2.2116 9.43 2.2439 9.73 2.2752 8.84 2.1793 9.14 2.2127 9-44 2.2450 9.74 2.2762 8.85 2.1804 915 2.2138 9.45 2.2460 9.75 2.2773 8.86 2.1815 9.16 2.2148 9.46 2.2471 9.76 2.2783 8.87 2.1827 9.17 2.2159 9.47 2.2481 9-77 2.2793 8.88 2.1838 9.18 •2.2170 9.48 2.2492 9.78 2.2803 8.89 2.1849 9.19 2.2181 9.49 2.2502 9.79 2.2814 8.90 2.1861 9.20 2.219^ 9.50 2.2513 9.80 2.2824 8.91 2.1872 9.21 2.2203 9.51 2.2523 9.81 2.2834 8.92 2.1883 9.22 2.2214 9.52 2.2534 9.82 2.2844 8-93 2. 1 894 923 2.2225 9.53 2.2544 9.83 2.2854 8.94 2. 1 905 9.24 2.2235 9.54 2.2555 9.84 2.2865 8.95 2.1917 9.25 2.2246 , 9.55 2.2565 9.85 2.2875 8.96 2.1928 9.26 2.2257 9.56 2.2576 9.86 2.2885 8.97 2.1939 9.27 2.2268 9.57 2.2586 9.87 2.2895 8.98 2.1950 9.28 2.2279 9.58 2.2597 9.88 2.2905 8.99 2.1961 9.29 2.2289 9-59 2.2607 9.89 2.2915 9.00 2.1972 9-30 2.2300 ' 9.60 2.2618 9.90 2.2925 9.01 2.1983 9-31 2.23II 9.61 2.2628 9.91 2.2935 9.02 2.1994 932 2.2322 9.62 2.2638 9.92 2.2946 903 2.2006 9-33 2.2332 9.63 2.2649 9-93 2.2956 9.04 2.2017 9-34 2.2343 9.64 2.2659 9.94 2.2966 9.05 2.2028 9-35 2.2354 9.65 2.2670 9-95 2.2976 9.06 2.2039 9-36 2.2364 i 9.66 2.2680 9.96 2.2986 9.07 2.2050 9-37 2.2375 i 967 2.2690 9-97 2.2996 9.08 2.2061 938 2.2386 9.68 2.2701 9.98 2.3006 : 9.09 2.2072 9-39 2.2396 9.69 2.271I 9.99 2.3016 9.10 1 2.2083 9.40 2.2407 9.70 2.2721 10.00 2.3026 10.25 2.3279 12.75 2.5455 15.50 2.7408 1 j 21.0 3.0445 10.50 2.3513 13.00 2.5649 16.0 2.7726 22.0 3.O9II 10.75 2.3749 13.25 2.5840 16.5 2.8034 23.0 3.1355 11.00 2.3979 13-50 2.6027 17.0 2.8332 24.0 3.I781 11.25 2.4201 13.75 2.62II 17.5 2.8621 25.0 3.2189 11.50 2.4430 14.00 2.6391 18.0 2.8904 26.0 3.2581 "75 2.4636 14.25 2.6567 18.5 2.9173 : 27.0 3.2958 12.00 2.4849 14.50 2.6740 19.0 2.9444 28.0 3.3322 12.25 2.5052 14.75 2.6913 19.5 2.9703 29.0 33673 12.50 2.5262 15.00 2.7081 20.0 2.9957 30.0 3.4012 MATHEMATICAL , TABLES. [IL— NUMBERS, OR DIAMETERS OF CIRCLES, CIR- ENCES, AREAS, SQUARES, CUBES, SQUARE ROOTS, BE ROOTS. 0.7854 3- 14 7.07 12-57 19.63 . 28.^7 38.48 . S0.26 63.61 78.54 95-03 .113.09 132-73 ■•53-93 173-71 .aoi.o6 226.98 .254.46 28J.52 .3 14- 15 346-36 ,380.13 415-47 452.38 490,87 .530-02 57^-55 615-75 660.52 .706.85 754-76 855-29 .907.92 962.11 :oi7.87 .075.21 13411 .. 256 289 I 1.331 ,.1,728 2,197 ! .. 2,744 3-375 ,.4,096 4-913 .- 5,832 6,859 ■9:261 15,62s 17,576 19,683 29,791 3^,768 35,937 39,3°4 42,875 46,656 50,653 54,872 59,3 '9 64,000 68,921 74,088 3-162 3-316 3-464 3.605 5-741 3.872 4.000 4-123 4.242 4-358 4-472 4-582 4-795 4.898 5.000 5-099 5-196 5-291 5-385 S-477 5-567 5.656 5-744 5-830 5.916 6.000 6.082 6.164 6.244 6.324 6.403 6.480 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 67 i? Nttjnber, Circum- CtroUar Sauaie. Cube. Square .Cube Diameter. ference. Area. ^^«4 t*a«a^«« Root. Root 43 135-oS 1452.20 1,849 79,507 6.557 3.503 44 138.23 ... 1520.52 ... 1,936 85,184 6.633 3.530 45 141.37 1590.43 2,025 91,125 6.708 3.556 46 144.51 ... 1661.90 ... 2,116 97,336 6.782 3.583 47 147.65 1734.94 2,209 103,823 6.855 3.608 48 150-79 ... 1809.55 ... 2,304 110,592 6.928 3-634 49 153.93 1885.74 2,401 117,649 7.000 3.659 50 157.08 ... 1963.49 ... 2,500 125,000 7.071 3.684 51 160.22 2042.82 2,6oi 132,651 7.I4I 3.708 52 163.36 ... 2123.71 ..: 2,704 140,608 7. 211 3.732 53 166.50 2206.18 2,809 148,877 7.280 3.756 54 169.64 ... 2290.21 ... 2,916 157,464 7.348 3.779 55 172.78 2375.82 3»025 166,375 7.416 3.802 56 17592 ... 2463.09 ... 3*136 I75»6i6 7.483 3.825 57 179.07 2551.75 3i249 185,193 7-549 3.848 58 182.21 ... 2642.08 ... 3*364 i95,"2 7.61S 3.870 59 185-35 2733-97 3,481 205,379 7.681 3.892 60 188.49 ... 2827.43 ... 3,600 216,000 7.745 3.914 61 19^-63 2922.46 3,721 226,981 7.810 3.936 62 194.77 ... 3019.07 ... 3,844 238,328 7.874 3.957 63 197.92 3117.24 3,969 250,047 7.937 3.979 64 201.06 ... 3216.99 ... 4^096 262,144 8.000 4.000 65 204.20 3318.30 4,225 274,625 8.062 4.020 66 207.34 ... 3421.18 ... 4,356 287,496 8.124 4.041 67 210.48 3525.65 4,489 300,763 8.185 4.061 68 213.62 ... 3631.68 ... 4,624 314,432 8.246 4.081 69 216.77 3739.28 4,761 328,509 8.306 4.IOI 70 219.91 ... 3848.45 ... 4,900 343,000 8.366 4. 12 I 71 223.05 3959.19 5,041 357,911 8.426 4.140 72 226.19 ... 4071.50 ... 5,184 373,248 8.485 4.160 73 22g.Z2» 4185.38 5,329 389,017 ^.544 4.179 . 74 ^S^Al ... 4300.84 ... 5,476 405,224 8.602 4.198 1 75 235.61 4417.86 5,625 421,875 8.660 4.217 76 238.76 ... 4536.45 ••• 5,776 438,976 8.717 4.235 77 241.90 4656.62 5,929 456,533 8.744 4.254 78 1 245-04 ... 4778.36 ... 6,084 474,552 8.831 4.272 79 ' 248.18 4901.66 6,241 493,039 8.888 4.290 1 W ^ 80 251-32 ... 5026.54 ... 6,400 512,000 8.944 4.308 81 ' 25446 5153.00 6,561 531,441 9.000 4.326 82 257.61 ... 5281.01 ... 6,724 551,368 9.055 4.344 83 260.75 5410.59 6,889 571,787 9. 1 10 4.362 84 ^es-^9 ... 5541.77 ... 7,056 592,704 9.165 4.379 85 %6 267.03 5674.50 7,225 614,125 9.219 4.396 270-17 ... 5808.80 ... 7,396 636,056 9.273 4.414 87 273-3^ 5944.67 7,569 658,503 9.327 4.431 88 276.46 ... 6082.11 ... 7,744 681,472 9.380 4.447 89 90 279.60 6221.13 7,921 704,969 9.433 4.461 282.74 ... 6361.72 ... 8,100 729,000 9.486 4.481 " ill 68 MATHEMATICAL TABLES. Number, 1 or Circum- Circular Sfluare. Cube. Square Cube Diameter. ference. Area. ^^%»A#^»« Root. Root. 91 285.88 6503.87 8,281 753*571 9.539 4.497 92 289.02 ... 6647.61 ... 8,464 778,688 9591 4.514 93 292.16 6792.90 8,649 804,357 9.643 4.530 94 295-31 ... 6939.78 ... 8,836 830,584 9.695 4.546 95 298.45 7088.21 9>o25 857*375 9.746 4.562 96 301-59 ... 7238.23 ... 9,216 884,736 9.797 4.578 97 304.73 7389.81 9,409 912,673 9.848 4.594 98 307.87 ... 7542.96 ... 9,604 941,192 9.899 4.610 99 3II.OI 7697.68 9,801 970,299 9.949 4.626 100 314.15 -. 7853.97 ...10,000 ... 1,000,000 10.000 4.641 lOI 317.30 8011.86 10,201 1*030,301 10.049 4.657 102 320.41 ... 8171.30 ...10,404 ... 1,061,208 10.099 4.672 103 323.58 8332.30 10,609 1,092,727 10.148 4.687 104 326.72 ... 8494.88 ...10,816 .,.. 1,124,864 10.198 4.702 105 329.86 8659.03 11,025 1*157,625 10.246 4.717 106 333.00 ... 8824.75 !... 11,236 ... 1,191,016 10.295 4.732 107 336.15 8992.04 11,449 1,225,043 10.344 4.747 108 339.29 ... 9160.90 ...11,664 ... 1,259,712 10.392 4.762 109 342.43 9331.33 11,881 1,295,029 10.440 4.776 no 345-57 - 9503.34 ...12,100 ■•• 1*331,000 10.488 4.791 III 348.71 9676.91 12,321 1,367.631 10.535 4.805 112 351.85 ... 9852.05 ...12,544 ... 1,404,928 10.583 4.820 113 355.01 10028.77 12,769 1,442,897 10.630 4.834 114 358.14 ...10207.05 ...12,996 ••• 1,481,544 10.677 4.848 115 361.28 10386.91 13*225 1,520,875 10.723 4.862 116 364.42 ...10568.34 ... 13*456 ... 1,560,896 10.770 4.876 117 367.56 10751.34 13,689 1,601,613 10.816 4.890 118 370.70 ...10935.90 ...13,924 ••• 1,643,032 10.862 4.904 119 373.81 III22.O4 14,161 1*685,159 10.908 4.918 120 376.99 ...11309.76 ...14,400 ... 1,728,000 10.954 4.932 121 380.1^ 11499.04 14,641 1,771,561 11.000 4.946 122 383.27 ...11689.89 ...14,884 ... 1,815,848 11.045 4.959 123 386.41 I1882.3I 15*129 1,860,867 11.090 4.973 124 389.55 ...12076.31 ...15*376 ... 1,906,624 11.135 4.986 125 392.70 12271.87 15*625 1*953,125 II. 180 5.000 126 395.84 ...12469.01 ...15*876 ... 2,000,376 11.224 5013 127 398.98 12667.71 16,129 2,048,383 11.269 5.026 128 402.12 ...12867.99 ...16,384 ... 2,097,152 11.313 5.039 129 405.26 13069.84 16,641 2,146,689 11.357 5.052 130 408.10 ...13273.26 ...16,900 ... 2,197,000 1 1. 401 5.065 131 411.54 13478.24 17,161 2,248,091 11.445 5.078 132 414.69 ...13694.80 ...17,424 ... 2,299,968 11.489 5.091 133 417.83 13892.94 17,689 2,352,637 ".532 5.104 134 420.97 ...14102.64 ...17,956 ... 2,406,104 .11.575 5."7 135 424.11 I4313.9I 18,225 2,460,375 II. 618 5.129 136 427.25 ...14526.75 ...18,496 ... 2,515,456 II. 661 5.142 137 430.39 I474I.I7 18,769 2*571,353 11.704 5-155 138 1 433.54 ••.14957.15 ...19,044 ... 2,620,872 11.747 5.167 NUMBERS, OR DIAMETERS OF CIRCLES, ftc. 69 Nanaber, nr Grcnm- Circular Square. Cube. Square Cube EKaxneter. f«rence. . Area. Root. Root. 139 436.68 I5174.7I 19,321 2,685,619 11.789 5.180 140 439.82 •••15393.84 ...19,600 ... 2,744,000 11.832 5-192 141 442.96 15614-53 19,881 2,803,221 11.874 5.204 142 446.10 ...15836.80 ...20,164 ... 2,863,288 11.916 5-217 143 449.24 16060.64 20,449 2,924,207 11.958 5.229 144 452.39 ...16286.05 ...20,736 ••• 2,985,984 12.000 5-241 145 455.53 16513.03 21,025 3,048,625 12.041 5-253 146 45».67 ...16741.58 ...21,316 ... 3,112,136 12.083 5.265 147 461.81 16971.70 21,609 3,176,523 12.124 5-277 148 464-95 ...17203.40 ...21,904 ••• 3,241,792 12.165 5.289 149 468.09 17436.66 22,201 3,307,949 12.206 5.301 150 471.24 ...17671.50 ...22,500 ••• 3,375,000 12.247 5.313 151 474.3S 17907.90 22,8oi 3,442,951 12.288 5.325 152 477.52 ...18145.88 ...23,104 ... 3,511,808 12.328 5-336 153 480.66 18385.42 23,409 3,581,577 12.369 5.348 154 483.80 ...18626.54 ...23,716 ... 3,652,264 12.409 5.360 155 , 486.94 18869.23 24,025 3,723,875 12.449 5.371 156 490.08 ...19113.49 ••.24,336 ... 3,796,416 12.489 5.383 157 493.23 1935932 24,649 3,869,893 12.529 5.394 158 496.37 ...19606.72 ...24,964 ... 3,944,312 12.569 5.406 159 499.51 19855.69 25,281 4,019,679 12.609 5.417 160 502.65 ...20106.24 ...25,600 ... 4,096,000 12.649 5.428 161 505-79 20358.35 25,921 4,173,281 12.688 5-440 162 508.93 ...20612.03 ...26,244 ... 4,251,528 12.727 5.451 163 512.08 20867.20 26,569 4,330,747 12.767 5.462 . 164 515.22 ...21 124. 1 1 ...26,896 ... 4,410,944 12.806 5-473 165 518.36 21382.51 27,225 4,492,125 12.845 S-484 166 521.50 ...21642.48 ...27,556 ••• 4,574,296 12.884 5.495 167 524-64 21904.02 27,889 4,657,463 12.922 5-506 168 527.78 ...22167.12 ...28,224 ••• 4,741,632 ^12.961 5-517 169 530-93 22431.80 28,561 4,826,809 13.000 5.528 170 1 534-07 ...22698.06 ...28,900 ••• 4,913,000 13.038 5.539 171 537.31 22965.88 29,241 5,000,211 13.076 5.550 172 540.35 •. 2323527 ...29,584 ... 5,088,448 13.114 5.561 173 543.49 23506.23 29,929 5,177,717 13.152 5.572 174 546.03 ...23778.77 ...30,276 ... 5,268,024 13.190 5.582 175 549. 78 . 24052.87 30,625 5,359,375 13.228 5.593 176 1 552.92 ...24328.55 ...30,976 ••• 5,451,776 13.266 5.604 177 556.06 24605.79 31,329 5,545,233 13.304 5.614 178 559-20 ...24884.61 ...31,684 - 5,639,752 13-341 5.625 179 562-34 25165.00 32,041 5,735,339 13-379 5.635 zSo 565.48 ...25446,96 ...32,400 ... 5,832,000 13.416 5.646 idi 568.62 25730.48 32,761 5,929,741 13453 5.656 182 571.77 ...26015.58 ..•33,124 ... 6,028,568 13.490 5.667 183 ; 574.91 26302.26 33,489 6,128,487 13^527 5.677 184 573.05 ...26590.50 ...33,856 ... 6,229,504 13-564 5.687 . 1S5 581.19 26880.31 34,225 6,331,625 13.601 5.698 186 584.33 ...27171.69 ••34,596 ... 6,434,856 13.638 5.708 70 MATHEMATICAL TABLES. Number, or Circum- Cirailar Square. Cube. Square Cube Diameter. ference. Area. Root. Root. 187 537.47 27464.65 34,969 6,539,203. 13.674 5.718 188 590.62 ..•27759-17 •-35>344 ... 6,644,672 I3.71I 5.728 189 593-76 28055.27 35,721 6,751,269 13-747 5.738 190 596.90 ...28352.94 ...36,100 ... 6,859,000 13.784 5.748 191 600.04 28652.17 36,481 6,967,871 13.820 5.758 192 603.18 ...28952.98 ...36,864 ... 7,077,888 13.856 5.768 193 606.32 29255.36 37,249 7,189,057 13.892 5.778 194 * 609.47 -29559.31 ...37,636 ... 7,301,384 13.928 5.788 195 612.61 29864.83 38,025 7,414,875 13.964 5.798 196 615.75 ...30171.92 ...38,416 - 7,529,536 14.000 5.808 197 618.89 30480.60 38,809 7,645,373 14.035 5.818 198 622.03 ...30790.82 ...39,204 ... 7,762,392 14.071 5.828 199 625.17 31102.52 39,601 7,880,599 14.106 5.838 200 628.32 ...31416.00 ...40,000 ... 8,000,000 14.142 5.848 201 631.46 31730.94 40,401 8,120,601 14.177 5.857 202 634.60 ...32047.46 ...40,804 ... 8,242,408 14.212 5.867 203 637.74 32365.54 41,209 8,365,427 14.247 5.877 204 640.88 ...32685.20 ...41,616 ... 8,489,664 14.282 5-886 205 644.02 33006.43 42,025 8,615,125 ' 14.317 5.896 206 647.16 •33329.23 ...42,436 ... 8,741,816 14.352 5.905 207 650-31 33653.60 42,849 8,869,743 14-387 5-915 208 653-45 -33979.54 ...43,264 ... 8,998,912 14.422 5-924 209 656.59 34307.05 43,681 9,123,329 14.456 5.934 210 659-73 ...34636.14 ...44,100 ... 9,261,000 14.491 5.943 211 662.87 34966.79 44,521 9,393,931 14.525 5.953 212 666.01 ...35299.01 -••44,944 ... 9,528,128 14.560 5.962 ^13 669.16 35632.81 45,369 9,663,597 14.594 5.972 214 672.30 ...35968.17 ...45,796 ... 9,800,344 14.628 5.981 215 675-44 36305.11 46,225 9,938,375 14.662 5.990 216 678.5S ...36643.62 ...46,656 ...10,077,696 14.696 6.000 217 681.71 36983.70 47,089 10,218,313 14.730 6.009 218 684.86 ••.37325.34 ..•47,524 ...10,360,232 14.764 6.018 219 688.01 37668.56 . 47,961 10,503,459 14.798 6.027 220 691.15 ...38013.36 ...48,400 ...10,648,000 14.832 6.036 221 694.29 38359-72 48,841 10,793,861 14.866 6.045 222 697.43 ...38707.65 ...49,284 ...10,941,048 14.899 6.055 223 700.57 39037.51 49,729 11,089,567 14.933 6.064 224 703-71 ...39408.23 ...50,176 ...11,239,424 14.966 6.073 225 706.86 39760.87 50,625 11,390,625 15.000 6.082 226 710.00 ...40115.09 ...51,076 --•",543,176 15-033 6.091 227 713.14 40470.87 51,529 11,697,083 15.066 6.100 228 716.28 ...40828.23 ...51,984 ••.11,852,352 15.099 6.109 229 719.42 41187.16 52,441 12,008,989 15-132 6. 1 18 230 722.56 ...41547.66 ...52,900 ...12,167,000 15-165 6.126 231 725.70 41909.72 53,361 12,326,391 15.198 6.135 232 728.85 ...42273.36 -..53,824 ...12,487,168 15.231 6.144 233 731.99 • 42638.58 54,289 12,649,337 15.264 6.^53 234 735.13 ...43005.36 ..-54,756 ...12,812,904 15.297 6.162 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 71 NnTTiber, or Circum- Circular Square. Cube. Square Cube Dumeter ference. • 1 Area. Root. Root. 235 i 738.27 43373-71 55,225 12,977,875 15329 6.171 236 741.41 ...43743-63 ...55,696 ...13,144,256 15.362 6.179 237 744-55 44115.11 56,169 13,312,053 15.394 6.188 23S 747.68 ...44488.19 ...56,644 ...13,481,272 15.427 6.197 239 750.88 44862.83 57,121 13,651,919 15-459 6.205 240 753.98 ...45239.04 ...57,600 ...13,824,000 I549I 6.214 241 757.12 45616.81 58,081 13,997,521 15.524 6.223 242 760.26 ...45996.16 ...58,564 ...14,172,488 15.556 6.231 243 763.40 46377.08 59,049 14,348,907 15.588 6.240 244 766.52 ...46759.57 ...59,536 ...14,526,784 15.620 6.248 245 769.92 47143-63 60,025 14,706,125 15^652 6.257 246 ' 772.83 ...47529.26 ...60,516 ...14,886,936 15.684 6.265 247 775-97 47916.46 61,009 15,069,223 15.716 6.274 248 779.11 ...48305.24 ...61,504 ...15,252,992 15.748 6.282 249 782.25 48695.58 62,001 15,438,249 15.779 6.291 250 785.40 ...49087.50 ...62,500 ...15,625,000 I5.81I 6.299 251 788.54 49480.98 63,001 15,813,251 15.842 6.307 252 791.68 ...49876.04 ...63,504 ...16,003,008 15.874 6.316 253 794.82 50272.66 64,009 16,194,277 15.905 6.324 254 797.96 ...50670.86 ...64,516 ...16,387,064 15.937 6.333 255 801.10 51070.63 65,025 16,581,375 15.968 6.341 256 804.24 ...51471-96 ...65,536 ...16,777,216 16.000 6.349 257 807.39 51874.88 66,049 16,974,593 16.031 6.357 258 810.53 ...52279.36 ...66,564 ••.17,173,512 16.062 6.366 259 813.67 52685.41 67,081 17,373,979 16.093 6.374 2€o 816.81 ...53093.04 ...67,600 •..17,576,000 16.124 6.382 , 261 819.95 53502.23 68,121 17,779,581 16.155 6.390 j 262 823.09 ...53912.99 ...68,644 ...17,984,728 16.186 6.398 263 826.24 54325.33 69,169 18,191,447 16.217 6.406 264 829.38 -54739.23 ...69,696 •••18,399,744 16.248 6.415 265 832.52 55154.71 70,225 18,609,625 f6.278 6.423 266 835.66 ...55571.76 ...70,756 ...18,821,096 16.309 6.431 267 838.80 55990.38 71,289 19,034,163 16.340 6.439 268 841.94 ...56410.56 ...71,824 ...19,248,832 16.370 6.447 269 845-09 56832.32 72,361 19,465,109 16.401 6.455 270 848.23 ...57255.66 ...72,900 ...19,683,000 16.431 6.463 271 851.37 57680.56 73,441 19,902,511 16.462 6.471 272 854-51 ...58107.03 ...73,984 ...20,123,648 16.492 6.479 . ^^^ 857.65 58535.07 74,529 20,346,417 16.522 6.487 1 274 860.79 ...58964.69 ...75,076 ...20,570,824 16.552 6.495 275 863.94 59393.87 75,625 20,796,875 16.583 6.502 276 ; 867.08 ...59828.63 ...76,176 ...21,024,576 16.613 6.510 277 870.22 60262.95 76,729 21,253,933 16.643 6.518 278 873.36 ...60698.85 ...77,284 ...21,484,952 16.673 6.526 279 876.50 61136.32 77,841 21,717,639 16.703 6.534 280 879.64 ...61573.36 ...78,400 ...21,952,000 16.733 6.542 281 882.78 62015.96 78,961 22,188,041 16.763 6.549 282 885.93 ...62458.14 ...79,524 ...22,425,768 16.792 6.557 / / / \ 72 MATHEMATICAL TABLES. Number. A « «• AAA l^^^s J or Circum- Circular Square. Cube. Square Cube Diameter. ference. Area. Root. Root. 283 889.07 62901.90 80,089 22,665,187 16.822 6.565 284 892.21 ...63347.22 ...80,656 ...22,906,304 16.852 6.573 285 895-35 63794.11 81,225 23,149,125 16.881 6.580 286 898.49 ...64242.57 ...81,796 •••23,393,656 16.9II 6.588 287 901.63 64692.61 82,369 23,639,903 16.941 6.596 288 904.78 ...65144.21 ...82,944 ...23,887,872 16.970 6.603 289 907.92 65597-39 83,521 24,137,569 17.000 6.611 290 911.06 ...66052.14 ...84,100 ...24,389,000 17.029 6.619 291 914.20 66508.45 84,681 24,642,171 17.059 6.627 292 91734 ...66966.34 ...85,264 ...24,897,088 17.088 6.634 293 920.48 67425.80 85,849 25*153,757 I7.II7 6.642 294 923.63 ...67886.83 ...86,436 ...25,412,184 17.146 6.649 295 926.77 68349.43 87,025 25*672,375 17.176 6.657 296 929.91 ...68813.60 ...87,616 ...25,934,336 17.205 6.664 297 933-05 69279.34 88,209 26,198,073 17.234 6.672 298 936.19 ...69746.66 ...88,804 ...26,463,592 17.263 6.679 299 939-33 . 70215.54 89,401 26,730,899 17.292 6.687 300 942.48 ...70686.00 ...90,000 ...27,000,000 17.320 6.694 301 945.62 71158.02 90,601 27,270,901 17.349 6.702 302 948.76 ...71631.62 ...91,204 ...27,543,608 17.378 6.709 303 951-90 72106.78 91,809 27,818,127 17.407 6.717 304 955-04 -••72583.52 ...92,416 ...28,094,464 17.436 6.724 305 958.18 73061.83 93,025 28,372,625 17.464 6.731 306 961.32 ..•73541.71 •93,636 ...28,652,616 17.493 6.739 307 964.47 74023.16 94,249 28,934,443 ^7.521 6.746 308 967.61 ...74506.18 ...94,864 ...29,218,112 17.549 6.753 309 970.75 74990.77 95,481 29,503,629 17-578 6.761 310 973-89 ...75476.94 ...96,100 ...29,791,000 17.607 6.768 311 977-03 75964.67 96,721 30,080,231 17.635 6.775 312 980. 1 7 ...76453.93 ...97,344 .. 30,371*328 17.663 6.782 313 983.32 76944-85 97,969 30,664,297 17.692 6.789 314 986.45 ...77437.29 ...98,596 ...30,959,144 17.720 6.797 315 989.60 77931.31 99,225 31,255,875 17.748 6.804 316 992.74 ...78426.89 ...99,856 .. .31*554,496 17.776 6.81 1 317 995-88 78924.06 100,489 31*855,013 17.804 6.818 318 999.02 ...79422.78 101,124 •..32,157,432 17.832 6.826 319 1002.17 79923.08 101,761 32,461,759 17.860 6.833 320 1005.31 ...80424.96 102,400 ...32,768,000 17.888 6.839 321 1008.45 80928.40 103,041 33,076,161 17.916 6.847 322 1011.59 ...81433.41 103,684 ...33*386,248 17-944 6.854 323 1014.73 81939.99 104,329 33,698,267 17.97^2 6.861 324 1017.47 ...82448.15 104,976 ...34,012,224 18.000 6.868 325 1021.02 82957.87 105,625 34,328,125 18.028 6.875 326 1024.16 ...83469.17 106,276 ..•34,645*976 18.055 6.882 327 1027.30 83982.60 106,929 34,965*783 18.083 6.889 328 1030.44 ...84496.47 107,584 ...35*287,552 18.IH 6.896 329 1033.58 85012.48 108,241 35,611,289 18.138 6.903 330 1036.72 ...85530.06 108,900 •-.35,937*000 18.166 6.910 NUMBERS, OR DIAMETERS OF CIRCLES, Ac. 73 Nomber, or Grcum- Circular Square. Cube. Square Cube Diameter. ference. Area. Root. Root. 331 1039.86 86049.20 109,561 36,264,691 18.193 6.917 332 1043.01 ...86569.92 110,224 ...36,594,368 18.221 6.924 333 1046.15 87092.22 110,889 36,926,037 18.248 6.931 334 1049.29 ...87616.08 111,556 ..•37,259,704 18.276 6.938 335 1052.43 88141.51 112,225 37,595,375 18.303 6.945 336 1055.57 ...88668.51 112,896 ...37^33,056 18.330 6.952 337 1058.71 89197.09 113,569 38,272,753 18.357 6.959 338 1061.86 ...89727.23 114,244 ...38,614,472 18.385 6.966 339 1065.02 90258.9s 114,921 38,958,219 18.412 6.973 340 1068.14 ...90792.24 115,600 ...39,304,000 18.439 6.979 341 1071.28 91327.09 116,281 39,651,821 18.466 6.986 342 1074.27 ■..91863.52 116,964 ...40,001,688 18.493 6.993 343 1077.56 92401.15 117,649 40,353,607 18.520 7.000 344 1080.71 ...92941.09 118,336 ...40,707,584 18.547 7.007 345 1083.85 93482.23 119,025 41,063,625 18.574 7.014 346 1086.99 ...94024.94 119,716 ...41,421,736 18.601 7.020 347 1090.35 94569.22 120,409 41,781,923 18.628 7.027 348 1093.07 ...95115.08 121,104 ...42.144,192 18.655 7.034 349 1096.41 95662.50 I2I,8oi 42,508,549 18.681 7.040 350 1099.56 ...96211.50 122,500 ...42,875,000 18.708 7.047 351 1102.70 96762.06 123,201 43,243,551 18.735 7.054 352 1105.84 ...97314.20 123,904 ...43,614,208 18.762 7.061 353 1 108.98 97867.90 124,609 43,986,977 18.788 7.067 354 III2.62 ...98423.18 125,316 ...44,361,864 18.815 7.074 355 III5.26 98980.03 126,025 44,738,875 18.842 7.081 356 1 118.40 ..•99538.45 126,736 ...45,118,016 18.868 7.087 357 II2I.55 100098.43 127,449 45,499,293 18.894 7.094 358 1124.69 100660.00 128,164 ...45,882,712 18.921 7.101 359 1127.83 101223.13 128,881 46,268,279 18.947 7.107 360 1130.97 101787.84 129,600 ...46,656,000 18.974 7.114 361 II34.II 102354. H 130,321 47,045,881 19.000 7.120 362 , "37.25 102921.95 131,044 .••47,437,928 19.026 7.127 3^3 1140.40 103491.31 131,769 47,832,147 19.052 7.133 364 1143.54 104062.35 132,496 ...48,228,544 19.079 7.140 365 1146.68 104634.91 133,225 48,627,125 19.105 7.146 366 1 149.82 105209.04 133,956 ...49,027,896 19.131 7.153 367 1152.96 10578474 134,689 49,430,863 19.157 7.159 368 1 156.10 106362.00 135,424 •..49,836,032 19.183 7.166 369 1159.25 106940.84 136,161 50,243,409 19.209 7.172 370 1162.39 107521.26 136,900 .-.50,653,000 19.235 7.179 371 1165.53 108103.22 137,641 51,064,811 19.261 7.185 372 1168.67 108686.79 138,384 ...51,478,848 19.287 7.192 373 II7I.81 IO9271.91 139,129 51,895,117 19.313 7.198 t 374 1 17495 109858.62 139,876 ...52,313,624 19.339 7.205 375 II78.IO 110446.87 140,625 52,734,375 19.365 7. 211 376 1 181.24 IIIO36.71 141,376 ...53,157,376 19391 7.218 377 1 18438 III628.II 142,129 53,582,633 19.416 7.224 378 1187.52 II222I.O9 142,884 ...54,010,152 19.442 7-230 74 MATHEMATICAL TABLES. i Number, or Circum-b Circular Square. Cube. Square ( iTube Diameter. ference. Arcx Root. I looL 379 1190.66 II2815.64 143,641 54,439,939 19.468 7. 237 380 1193.80 II34II.76 144,400 ...54,872,000 19.493 7 •243 381 1196.94 114009.46 145,161 55,306,341 19.519 7. 249 382 1200.09 114608.70 145*924 •..55,742,968 19-545 7' 256 383 1203.23 115209.54 146,689 56,181,887 19.570 7. ,262 384 1206.37 115811*94 147,456 ...56,623,104 19.596 7. 268 385 1209.51 I16415.9I 148,225 57,066,625 19.621 7. '275 386 1212.65 117021.45 148,996 ...57,512,456 19.647 7. .281 387 1215.79 117628.57 149,769 57,960,603 19.672 7 .287 388 1218.94 118237.25 150,544 ...58,411,072 19.698 7. .294 389 1222.08 I18846.51 151,321 58,863,869 19.723 7. 299 390 1225.22 119453.94 152,100 -••59,319,000 19.748 7 ■306 391 1228.36 120072.73 152,881 59,776,471 19.774 7. 312 392 1231.50 120687.70 153,664 ...60,236,288 19.799 7< 319 393 1234.64 121304.24 154,449 60,698,457 19.824 7, .325 394 1237.79 121922.43 155,236 ...61,162,984 19.849 7. 331 395 1240.93 122542.03 156,025 61,629,875 19-875 7- 337 396 1244.07 123163.28 156,816 ...62,099,136 19.899 7 343 397 1247.21 123786.10 157,609 62,570,773 19.925 7 ■349 398 1250.35 I24412.IO 158,404 •..63,044,792 19.949 7 .356 399 1253.49 125036.46 159,201 63,521,199 19.975 7 362 400 1256.64 125664.00 160,000 ...64,000,000 20.000 7 ■368 401 1259.78 126293.10 160,801 64,481,201 20.025 7. .374 402 1262.92 126923.88 161,604 ...64,964,808 20.049 7 .380 403 1266.06 127556.02 162,409 65,450,827 20.075 7. .386 404 1269.20 128189.84 163,216 ••65,939,264 20.099 7. •392 405 1272.34 128825.23 164,025 66,430,125 20.125 7 •399 406 1275.48 129462.19 164,836 ...66,923,416 20.149 7. 405 407 1278.63 I30IOO.71 165,649 67,419,143 20.174 7. .411 408 1281.77 130740.82 166,464 ...67,911,312 20.199 7 .417 409 1284.91 131382.49 167,281 68,417,929 1 20.224 7 .422 410 1288.05 132025.74 168,100 ...68,921,000 20.248 7. .429 411 I29I.19 132670.55 168,921 69,426,531 20.273 7. 434 412 1294.32 133316.93 169,744 ••-69,934,528 20.298 7. 441 413 1297.48 133964.89 170,569 70,444,997 20.322 7 ■447 414 1300.62 134614.41 171,396 -■•70,957,944 20.347 7 ■453 415 1303-76 135265.51 172,225 71,473,375 20.371 7. ■459 416 1306.90 I35918.18 173,056 ...71,991,296 20.396 7. 465 417 1310.04 136572.42 173,889 72,511,713 20.421 7. .471 418 I313.18 137228.22 174,724 • -73,034,632 20.445 7- 477 419 1316.32 137885.69 175,561 73,560,059 20.469 7. .483 420 1319-47 138544.56 176,400 ...74,088,000 20.494 7. 489 421 1322.61 139205.08 177,241 74,618,461 20.518 7 •495 422 1325.75 139867.17 178,084 ...75,151,448 20.543 7- .501 423 1328.89 140530.83 178,929 75,686,967 20.567 7 ■507 424 1332.03 I41196.07 179,776 ...76,225,024 20.591 7. •513 425 1335-18 141862.87 180,625 76,765,625 20.615 7 .518 426 1338.32 142531.25 181,476 •.■77,308,776 \ 20.639 7 ■524 NUMBERS, OR DIAMETERS OF CIRCLES, 4c. 75 Nnmber, or Grcum- Circular S<]uare. Cube. Square Cube Diameter. ference. Area. Root. Root. 427 1341.46 I432OI.I9 182,329 77,854,483 20.664 7.530 428 1344.60 143872.71 183,184 ...78,402,752 20.688 7.536 429 1347.74 144545.80 184,041 78,953,589 20.712 7.542 430 1550.88 145220.46 184,900 ...79,507,000 20.736 7.548 431 1354.02 145696.68 185,761 80,062,991 20.760 7.554 432 1357.17 14657448 186,624 ...80,621,568 20.785 7-559 433 1360.33 147253.85 187,489 81,182,737 20.809 7.565 434 1363.45 147934.80 188,356 ...81,746,504 20.833 7.571 435 1366.59 I48617.3I 189,225 . 82,312,875 20.857 7.577 436 1369.73 149301.39 190,096 ...82,881,856 20.881 7.583 437 1372.87 149987.05 190,969 83,453,453 20.904 7.588 438 1376.02 150674.27 191,844 ...84,027,672 20.928 7.594 439 1379.16 151362.87 192,721 84,604,519 20.952 7.600 440 1382.30 152053.44 193,600 ...85,184,000 20.976 7.606 441 1385.44 152745.37 194,481 85,766,121 21.000 7.612 442 1388.58 153438.88 195*364 .■.86,350,388 21.024 7.617 443 1391.72 154133.96 196,249 86,938,307 21.047 7.623 444 1394.87 154830.61 197,136 ...87,528,384 21.071 7.629 445 1398.01 155528.83 198,025 88,121,125 21.095 7.635 446 I4OI.I5 156228.62 198,916 ...88,716,536 21. 119 7.640 447 1404.29 156929.98 199,809 89,314,623 21.142 7.646 448 1407.43 157632.92 200,704 •••89,915,392 21.166 7.652 449 1410.57 158337.42 201,601 90,518,849 21.189 7.657 450 1413.72 159043.50 202,500 ...91,125,000 21.213 7.663 451 1416.86 I5975I.I4 203,401 91,733,851 21.237 7.669 452 1420.00 160460.36 204,304 ...92,345,408 21.260 7.674 453 1423.14 161171.14 205,209 92,959,677 21.284 7.680 454 1426.28 161883.50 206,106 •••93,576,664 21.307 7.686 455 1429.42 162597.43 207,025 94,196,375 21.331 7.691 456 1432.56 163312.93 207,936 ...94,818,816 21.354 7.697 ' 457 1435.71 164030.20 208,849 95,443,993 21.377 7703 45« 1438.85 164748.64 209,764 ...96,071,912 21.401 7.708 459 1441.99 165468.85 210,681 96,702,579 21.424 7.714 460 1445-13 166190.64 211,600 • -97,336,000 21.447 7.719 , 461 1448.27 166913.99 212,521 97,972,181 21.471 7.725 462 I45I.4I 167638.91 213,444 ...98,611,128 21.494 7.731 463 1454.56 168365.41 214,369 99,252,847 21.517 7.736 464 1457.70 169093.47 215,296 •••99,897,345 21.541 7.742 . 465 1460.84 169823.II 216,225 100,544,625 21.564 7.747 . 466 1463.98 17055432 2171I56 101,194,696 21.587 7.753 I 467 1467.12 I71287.IO 218,089 101,847,563 21.610 7.758 1 468 1470.26 172021.44 219,024 102,503,232 21.633 7.764 ! 469 1473.41 172757.36 219,961 103,161,709 21.656 7.769 470 1476.55 173494.86 220,900 103,823,000 21.679 7^775 471 1479.69 174233.92 221,841 104,487,111 21.702 7.780 i ^72 1482.83 17497455 222,784 105,154,048 21.725 7.786 473 1485.97 175716.75 223,729 105,823,817 21.749 7.791 474 1 489. 1 1 176460.45 224,676 166,496,424 21.771 7.797 1^ MATHEMATICAL TABLES. Number, or Circum- Circular Soimrc Cube. Square Cube Diameter. ference. Area. fcj*l >M1 i ^* Root. Root. 475 1492.26 177205.87 225,625 107,171,875 21.794 7.802 476 1495.36 177952.79 226,576 107,850,176 21.817 7.808 477 1498.54 178701.27 227,529 108,531,333 21.840 7.813 478 1501.68 179451.33 228,484 109,215,352 21.863 7.819 479 1504.82 180202.96 229,441 109,902,239 21.886 7.824 480 1507.96 180956.16 230,400 110,592,000 21.909 7.830 481 I5II.IO 181712.92 231,361 111,284,641 21.932 7.835 482 1514.25 182467.26 232,324 111,980,168 21.954 7.840 483 1517.39 183225.18 233,289 112,678,587 21.977 7.846 484 1520.53 183984.66 234,256 "3,379,904 22.000 7.851 485 1523.67 184745.71 235,225 114,084,125 22.023 7.857 486 1526.81 185508.33 236,196 114,791,256 22.045 7.862 487 1529.95 186272.53 237,169 115,501,303 22.069 7.868 488 1533.90 187038.29 238,144 116,214,272 22.091 7.873 489 1536.24 187805.63 239,121 116,936,169 22.113 7.878 490 1539.38 188574,54 240,100 117,649,000 22.136 7.884 491 1542.52 189345.01 241,081 118,370,771 22.158 7.889 492 1545.66 I9OII7.06 242,064 119,095,488 22.181 7.894 493 1548.80 190890.68 243,049 119,823,157 22.204 7.899 494 1551.95 191665.87 244,036 120,553,784 22.226 7.905 495 155509 192442.63 245,025 121,287,375 22.248 7.910 496 1558.23 193220.96 246,016 122,023,936 22.271 7.915 497 1561.37 194000.86 247,009 122,763,473 22.293 7.921 498 1564.51 194782.34 248,004 123,505,992 22.316 7.926 499 1567.55 195565.38 249,001 124,251,499 22.338 7.932 500 1570.80 196350.00 250,000 125,000,000 22.361 7.937 501 1573.94 I97136.18 251,001 125,751,501 22.383 7.942 502 1577.08 197923.94 252,004 126,506,008 22.405 7.947 503 1580.22 198713.26 253,009 127,263,527 22.428 7.953 504 1583.36 199504.16 254,016 128,024,864 22.449 7.958 505 1586.50 200296.63 255,025 128,787,625 22.472 7-963 506 1589.64 201090.67 256,036 129,554,216 22.494 7.969 507 1592.79 201886.28 257,049 130,323,843 22.517 7.974 508 1595.93 202683.46 258,064 131,096,512 22.539 7.979 509 1599.07 203487.70 259,081 131,872,229 22.561 7.984 510 1602.21 204282.54 260,100 132,651,000 22.583 7.989 5" 1605.35 205084.43 261,121 133,432,831 22.605 7.995 512 1608.49 205887.84 262,144 134,217,728 22.627 8.000 513 161I.64 206692.93 263,169 135,005,697 22.649 8.005 514 1614.78 207499.53 264,196 135,796,744 22.671 8.010 515 1617.92 208307.71 265,225 136,590,875 22.694 8.016 516 1621.06 209117.46 266,256 137,388,096 22.716 8.021 517 1624.20 209928.78 267,289 138,188,413 22.738 8.026 S18 1627.34 210741.66 268,324 138,991,832 22.759 8.031 519 1630.49 2II556.I2 269,361 139,798,359 22.782 8.036 520 1633.63 212372.16 270,400 140,608,000 22.803 8.041 521 1636.77 213189.76 271,441 141,420,761 22.825 8.047 522 1639.93 214008.93 272,484 142,236,648 22.847 8.052 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 77 ' Nnmber, or OrcoiB- Circular Square. Cube, Square Cube iDofDClCT. Area. ^0^mm^^0^ Root. Root. S»3 1643.05 214829.67 273,529 143,055,667 22.869 8.057 5«4 1646.19 215651.99 274,576 143,877,824 22.891 8.062 5»5 1649.34 216475-87 275,625 144,703,125 22.913 8.067 526 1652.48 217301.33 276,676 145,531,576 22.935 8.072 527 1655.62 218128.35 277,729 146,363,183 22.956 8.077 5*8 1658.76 218956.95 278,784 147,197,952 22.978 8.082 5*9 1661.90 219787.12 279,841 148,035,889 23.000 8.087 530 1665.04 220618.86 280,900 148,877,000 23.022 8.093 531 1668.18 221452.16 281,961 149,721,291 23.043 8.098 ' 53* 1671-33 222287.04 283,024 150,568,768 23.065 8.103 533 1674,47 223123.50 284,089 151,419,437 23-087 8.108 534 * 1677.61 223961.52 285,156 152,273,304 23.108 8.II3 535 1680.75 224801. II 286,225 153,130,375 23.130 8. 118 536 1683.80 225642.27 287,296 153,990,656 23.152 8.123 537 I 1687.04 226487.01 288,369 154,854,153 23-173 8.128 538 . 1690.18 2273293^ 289,444 155,720,872 23-195 8.133 539 1 1693.32 228175.19 290,521 156,590,819 23.216 8.138 540 1696.46 229022.64 291,600 157,464,000 23-238 8.143 541 1699.60 229871.65 292,681 158,340,421 23-259 8.148 542 1702.74 230722.24 293,764 159,220,088 23.281 8.153 543 1705.88 231574.40 294,849 160,103,007 23.302 8.158 544 1709.03 232428.13 295,936 160,989,184 23-324 8.163 545 1712.17 233283.43 297,025 161,878,625 23-345 8.168 546 ' 1715-31 234140.30 298,116 162,771,336 23367 8.173 ' 547 1718.45 234998.74 299,209 163,667,323 23-388 8.178 548 . 1721.59 235858.76 300,304 164,566,592 23-409 8.183 549 • 1724.73 236720.34 301,401 165,469,149 23-431 8.188 550 ' 1727.88 237583-50 302,500 166,375,000 23-452 8.193 551 1731.02 238448.22 303,601 167,284,151 23-473 8.198 55* 1734.16 239314.52 304,704 168,196,608 23495 8.203 ' 553 1737.30 240182.38 305,809 169,112,377 235^6 8.208 554 I 1740.44 241051.82 306,916 170,031,464 23-537 8.213 555 1743.58 241922.83 308,025 170,953,875 23-558 8.218 556 1746.72 242795.41 309,136 171,879,616 23-579 8.223 557 1749.77 243669.56 310,249 172,808,693 23.601 8.228 558 1753.09 244545.28 311,364 173,741,112 23.622 8.233 559 1756.^^5 245422.57 312,481 174,676,879 23-643 8.238 560 1759.29 246301.44 313,600 175,616,000 23.664 8.242 561 1762.43 247181.87 314,721 176,558,481 23.685 8.247 562 ! 1765.57 248063.87 315,844 177,504,328 23.706 8.252 563 1768.72 248947.45 316,969 178,453,547 23.728 8.257 564 1771.86 249832.59 318,096 179,406,144 23.749 8.262 565 1775.00 250719-31 319,225 180,362,125 23.769 8.267 566 1778.14 251607.60 320,356 181,321,496 23.791 8.272 567 1781.28 252497.36 321,489 182,284,263 23.812 8.277 568 1 1784.42 253388.88 322,624 183,250,432 23833 8.282 569 1787.57 254281.88 323,761 184,220,009 23-854 8.286 570 1 1790.71 255176.64 324,900 185,193,000 23-875 8.291 78 MATHEMATICAL TABLES. Number, or Circum- Circular Square. Cube. Square Cube Diameter. ference. Area. Root. Root. 571 1793-85 256072.60 326,041 186,169,411 23.896 8.296 572 1796.99 256970.31 327,184 187,149,248 23.916 8.301 573 1800.13 257869.59 328,329 188,132,517 23-937 8.306 574 1803.27 258770.45 329,476 189,119,224 23.958 8.3 II 575 1806.42 259672.87 330,625 190,109,375 23979 8.315 576 1809.56 260576.87 331,776 191,102,976 24.000 8.320 577 1812.80 261482.43 332,929 192,100,033 24.021 8.325 578 1815.84 262388.57 334,084 193,100,552 24.042 8.330 579 1818.98 263298.28 335,241 194,104,539 24.062 8.335 580 1822.12 264208.56 336,400 195,112,000 24,083 8.339 581 1825.26 265120.46 337,561 196,122,941 24.104 8.344 582 1828.41 266033.82 338,724 197,137,368 24.125 8.349 583 1831.55 266948.82 339,889 198,155,287 24.145 8.354 584 1834.69 267865.38 341,056 199,176,704 24.166 8.359 585 1837.83 268783.57 342,225 200,201,625 24,187 8.363 586 1840.97 269703.21 343,396 20i;23o,o56 24.207 8.368 587 1 844. 1 1 270624.49 344,569 202,262,003 24,228 8.373 588 1847.26 271547.33 345,744 203,297,472 24.249 8.378 589 1850.40 272471.75 346,921 204,336,469 24.269 8.382 590 1853-54 273397.74 348,100 205,379,000 24.289 8.387 591 1856.68 274325.29 349,281 206,425,071 1 24.310 8.392 592 1859.82 275254,42 350,464 207,474,688 24.331 8.397 593 1862.96 276185.12 351,649 208,527,857 24.351 8.401 594 1 866. 11 277117.39 352,836 209,584,584 24.372 8.406 595 1869.25 278051.23 354,025 210,644,875 24-393 8.4II 596 1872.39 278986.64 355,216 211,708,736 24.413 8.415 597 1875.53 279923.62 356,409 212,776,173 24.433 8.420 598 1878.67 280862.18 357,604 213,847,192 24.454 8.425 599 1881.81 281802.30 358,801 214,921,799 24.474 8.429 600 1884.96 282744.00 360,000 216,000,000 24.495 8.434 601 1888.10 283687.26 361,201 217,081,801 24.515 8.439 602 1891.24 284632.10 362,404 218,167,208 24.536 8.444 603 1894.38 285578.50 363,609 219,256,227 24.556 8.448 604 1897.52 286526.48 364,816 220,348,864 24.576 8.453 605 1900.66 287476.03 366,025 221,445,125 24.597 8.458 606 1903.80 288426.15 367,236 222,545,016 i 24.617 8.462 607 1906.95 289379.84 368,449 223,648,543 24.637 8.467 608 1910.09 290334.10 369,664 224,755,712 24.658 8.472 609 1913.23 291289.93 370,881 225,866,529 24.678 8.476 610 1916.37 2^)2247.34 372,100 226,981,000 24.698 8.481 611 1919.51 293206.31 373,321 228,099,131 24.718 8.485 612 1922.65 294166.85 374,544 229,220,928 24.739 8.490 613 1925.80 295128.97 375,769 230,346,397 24.758 8.495 614 1928.94 296092.65 376,996 231,475,544 24.779 8.499 615 1932.08 297057.91 378,225 232,608,375 24.799 8.504 616 1935-22 298024.74 379,456 233,744.896 ' 24.819 8.509 617 1938.36 298993.14 380,689 234,885,113 24839 8.513 618 1941.50 299963.00 381,924 236,029,032 1 24.859 8.518 NUMBERS, OR DIAMETERS OF CIRCLES, Sec 79 Nomber, Dbm«ter. CSrcuin- ferenc«. Circular Area. Square. Cube. Square Root. Cube Root. 6io 1 ^ 1944.65 300934,64 383,161 237,176,659 24.879 8.522 620 1947.79 301907.76 384,400 238,628,000 24.899 8.527 621 1 1950.93 302S82.44 3^5Mx 239,483,061 24.919 8-532 622 1954-07 303858.69 386,884 240,641,848 24.939 8.536 6^3 1957.21 304836.51 388,129 241,804,367 24.959 8.541 624 1960.35 305815.91 389,376 242,970,624 24.980 8.545 625 1963.50 306796.87 390,625 244,140,625 25.000 8.549 626 1966.64 307779.41 391,876 245,314,376 25.019 8.554 627 1969.78 308763.41 393,129 246,491,883 25.040 8.559 628 1972.92 309749.19 394,384 247,673,152 25-059 8.563 629 1976.06 310736.44 395,641 248,858,189 25.079 8.568 630 1979.20 311725.26 396,900 250,047,000 25.099 8.573 631 1982.34 312715.64 398,161 25^239,591 25.119 8.577 ' 632 1985.49 313707.58 399.424 252,435,968 25.139 8.582 ^3S 1988.63 31470I.I4 400,689 253,636,137 25.159 8.586 634 1991.77 315696.64 401,956 254,840,104 25-179 8.591 635 1994,91 316692.91 403,225 256,047,875 25.199 8.595 636 1998.05 31769I.15 404,496 257,259,456 25.219 8-599 637 2001.19 318690.97 405,769 258,474,853 25.239 8.604 638 2004.34 319692.35 407,044 259,694,072 25.259 8.609 639 2007.48 320695.31 408,321 260,917,119 25.278 8.613 640 2010.62 321699.84 409,600 262,144,000 25.298 8.6i8 641 2013.76 322705.93 410,881 263,374,721 25.318 8.622 642 2016.90 323713.60 412,164 264,609,288 25.338 8.627 643 2020.04 324722.84 413,449 265,847,707 25-357 8.631 644 2023.19 325733-65 414,736 267,089,984 25-377 8.636 645 2026.33 326746.03 416,025 268,836,125 25-397 8.640 646 2029.47 327759.98 417,316 269,586,136 25.416 8.644 647 2032.61 328775.50 418,609 270,840,023 25.436 8.649 648 2035.76 329792.60 419,904 272,097,792 25.456 8.653 649 2038.89 330811.26 421,201 273,359,449 25.475 8.658 650 2042.04 331831.50 422,500 274,625,000 25-495 8.662 1 651 2045.18 332853.40 423,801 275,894,451 25-515 8.667 652 2048.32 333876.68 425,104 277,167,808 25-534 8.671 653 2051.46 334901.62 426,409 278,445,077 25554 8.676 654 2054.60 335928.14 427,716 279,726,264 25-573 8.680 ' 655 2057.74 336956.23 429,025 281,011,375 25.593 8.684 656 2060.88 3379S5.89 430,336 282,800,416 25.612 8.689 657 2064.03 339017.12 431,649 283,593,393 25.632 8.693 658 2067.17 340049.92 432,964 284,890,312 25-651 8.698 659 2070.31 341084.29 434,281 286,191,179 25.671 8.702 660 2073.45 342120.24 435,600 287,496,000 25-690 8.706 661 . 2076.59 343157-75 436,921 288,804,781 1 25.710 8.711 662 2079.73 344196.33 438,244 290,117,528 25.720 8.715 663 2082.88 345237.49 439,569 291,434,247 1 25.749 8.719 664 2086.02 346279.71 440,896 292,754,944 25.768 8.724 665 2089.16 347323-51 442,225 294,079,625 25-787 8.728 666 2092.30 ^348368.88 443,556 295,408,296 25.807 8.733 8o MATHEMATICAL TABLES. Number, or Circum- Circular Sauare. Cube. Square Cube Diameter. ference. Area. M^^i^KA^hM ^r» Root. Root. 667 2095.44 349416.40 444,889 296,740,963 25.826 8.737 668 2098.58 350464.32 446,224 298,077,632 25.846 8.742 669 2101.73 351514-30 447,561 299,418,309 25.865 8.746 670 2104.87 352566.06 448,900 300,763,000 25.884 8.750 671 2108.01 353619.28 450,241 302,111,711 25.904 8.753 672 2III.15 354674.07 451,584 303,464,448 25-923 8.759 673 2114.29 355730-43 452,929 304,821,217 25.942 8.763 674 2117-43 356788-37 454,276 306,182,024 25.961 8.768 675 2i2a58 357847.87 455,625 307,546,875 25.981 8.772 676 2123.72 358908.95 456,976 308,915,776 26.000 8.776 677 2126.86 359971.59 458,329 310,288,733 26.019 8.781 678 2130.00 361035.81 459,684 311,665,752 26.038 8.785 679 2133-14 362101.60 461,041 313,046,839 26.058 8.789 680 2136.28 363168.96 462,400 314,432,000 26.077 8.794 681 2139.42 364237.88 463,761 315,821,241 26.096 8.798 682 2142.57 365308.38 465,124 317,214,568 26.115 8.802 683 2145.71 366380.40 466,489 318,611,987 26.134 8.807 684 2148.85 367454.10 467,856 320,013,504 26.153 8.81 1 685 2151.99 368529.31 469,225 321,419,125 26.172 8.815 686 2155-13 369600.60 470,596 322,828,856 26.192 8.819 687 2158.27 370684.45 471,969 324,242,703 26.211 8.824 688 2161.42 371764.37 473*344 325,660,672 26.229 8.828 689 2164.56 372845.87 474,721 327,082,769 26.249 8.832 690 2167.70 373928.94 476,100 328,509,000 26.268 8.836 1 691 2170.84 375013-57 477,481 329,939*371 26.287 8.841 692 2173.98 376099.78 478,864 331,373,888 26.306 8.845 693 2177.12 377187-56 480,249 332,812,557 26.325 8.849 694 2180.27 378276.91 481,636 334,255,384 26.344 8.853 695 2183.41 379367-83 483,025 335,702,375 26.363 8.858 696 2186.55 380460.32 484,416 337,153,536 26.382 8.862 697 2189.69 381554-38 485,809 338,608,873 26.401 8.866 698 2192.83 382650.02 487,204 340,068,392 26.419 8.870 699 2195.97 383747.22 488,601 341,532,099 26.439 8.875 700 2199.12 384846.00 490,000 343,000,000 26.457 8.879 701 2202.26 385949-52 491,401 344,472,101 26.476 8.883 702 2205.40 387048.26 492,804 345,948,088 26.495 8.887 703 2208.54 388151.74 494,209 347,428,927 26.514 8.892 704 2211.68 389256.80 495,616 348,913,664 26.533 8.896 705 2214.82 390363-43 497,025 350,402,625 26.552 8.900 yo6 2217.96 391471-63 498,436 351,895,816 26.571 8.904 707 2221. II 392581.40 499,849 353,393,243 26.589 8.908 708 2224.25 393692.74 501,264 354,894,912 26.608 8.913 709 2227.39 394805.65 502,681 356,400,829 26.627 8.917 710 2230.53 395920.14 504,100 357,911,000 26.644 8.921 711 2233,67 397036.19 505,521 359,425,431 26.664 8.925 712 2236.81 398151.81 506,944 360,944,128 26.683 8.929 713 2239.96 399273.01 508,369 362,467,097 26.702 8.934 714 2243.10 400393.73 509,796 363,994,344 26.721 8.938 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 8l I Kiunber, or Diameter. 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 Grcum- ference. 2246.24 2249.38 2252.52 2255.66 2258.81 2261.95 2265.09 226S.23 2271.37 2274-51 2277.66 2280.80 2283.94 2287.08 2290.22 2293.36 2296.50 2299.65 2302.79 2305.93 2309-07 2312.21 2315.35 2318.50 2321.64 2324.78 2327.92 2331.06 2334.20 2337.35 2340.49 2343.63 2346.77 2349.91 2353.05 2356.20 2359.34 2362.48 2365.62 2368.76 2371.90 2375.04 2378.19 2381.33 2384.47 2387.61 2390.75 2393-89 Grcular Area. 4OI516.II 402640,02 403765.50 404892.54 406021.16 407151.36 408283.32 409416.45 410551.25 411687.93 412825.87 413965.24 415106.06 416249.43 417393-76 418539.66 419687.12 420836.14 421986.78 423138.96 424292.71 425442.03 426604.93 427763-39 428923.43 430085.04 431248.21 432412.96 433579-28 434747.17 435916.63 437087.66 438260.26 439434-48 440610.18 441787.50 442966.38 444146.84 445328.86 446512.46 447697-63 448884.37 450072.68 451262.56 452454.01 453647.04 454841.63 456037.87 Square. 5ii>225 512,656 514,089 515*524 516,961 518,400 519,841 521,284 522,729 524>i76 525*625 527,076 528,529 529*984 531,441 532,900 534,361 535,824 537,289 538,756 540,225 541,696 543,169 544,644 546,121 547,600 549,081 550,564 552,049 553,536 555,025 556,516 558,009 559,504 561,001 562,500 564,001 565,504 567,009 568,516 570,025 571,536 573,049 574,564 576,081 577,600 579,^21 580,644 Cube. 365,525,875 367,061,696 368,601,813 370,146,232 371,694,959 373,248,000 374,805,361 376,367,048 377,933,067 379,503,424 381,078,125 382,657,176 384,240,583 385,828,352 387,420,489 389,017,000 390,617,891 392,223,168 393,832,837 395,446,904 397,065,375 398,688,256 400,315,553 401,947,272 403,583,419 405,224,000 406,869,021 408,518,488 410,172,407 411,830,784 413,493,625 415,160,936 416,832,723 418,508,992 420,189,749 421,875,000 423,564,751 424,525,900 426,957,777 428,661,064 430,368,875 432,081,216 433,798,093 435,519,512 437,245,479 438,976,000 440,711,081 442,450,728 Square Cube Root. Root. 26.739 8.942 26.758 8.946 26.777 8.950 26.795 8.954 26.814 8.959 26.833 8.963 26.851 8.967 26.870 8.971 26.889 8.975 26.907 8.979 26.926 ^.983 26.944 8.988 26.963 8.992 26.991 8.996 27.000 9.000 27.018 9.004 27.037 9.008 27.055 9.012 27.074 9.016 27.092 9.020 27. Ill 9-023 27.129 9.029 27.148 9-033 27.166 9-037 27.184 9.041 27.203 9-045 27.221 9.049 27.239 9-053 27.258 9057 27.276 9.061 27.295 9.065 27.313 9.069 27.331 9-073 27.349 9-077 27.368 9.081 27.386 9.086 27.404 9.089 27.423 9.094 27.441 9.098 27.459 .9.102 27.477 9.106 27.495 9.109 27-514 9.II4 27-532 9. 1 18 27.549 9.122 27.568 9.126 27.586 9.129 27.604 9-134 iV 82 MATHEMATICAL TABLES. Number, or Diameter. Circum- ference. Circular Area. Square. Cube. Square Root. Cube Root, 763 2397.04 457235.53 582,169 444,194,947 27.622 9.138 764 2400.18 458435.83 583,696 445,943,744 27.640 9.142 765 2403.32 459635.71 585*225 447,697,125 27.659 9.146 766 2406.46 460838.16 586,756 449,455,096 27.677 9.149 767 2409.60 462042.18 588,289 451,217,663 27.695 9.154 768 2412.74 463247.76 589,824 452,984,832 27.713 9.158 769 2415.98 464454.92 59i»36i 454,756,609 27.731 9.162 770 2419.03 465663.66 592,900 456,533,000 27.749 9.166 771 2422.17 466873.96 594,441 458,314,011 27.767 9.169 772 2425-31 468085.83 595,984 460,099,648 27.785 9.173 77^ 2428.45 469299.27 597,529 461,889,917 27.803 9.177 774 2431-59 470514.29 599,076 463,684,824 27.821 9.181 775 2434-74 471730.87 600,625 465,484,375 27.839 9.185 776 2437.88 47^2949.03 602,176 467,288,576 27.857 9.189 777 2441.02 474168.75 603,729 469,097,433 27.875 9.193 778 2444.16 475396.05 605,284 470,910,952 27.893 9.197 779 2447.30 476612.92 606,841 472,729,139 27.910 9.201 780 2450.44 477837.36 608,400 474,552,000 27.928 9.205 781 2453.58 479063.36 609,961 476,379,541 27.946 9.209 782 2456.73 480290.94 611,524 478,211,768 27.964 9.213 783 2459.87 481520.10 613,089 480,048,687 27.982 9.217 784 2463.01 482750.82 614,656 481,890,304 28.000 9.221 785 2466.15 483983.11 616,225 483,736,025 28.017 9.225 786 2469.29 485216.97 617,796 485,587,656 28.036 9.229 787 2472.43 486452.41 619,369 487,443,403 28.053 9.233 788 2475.48 487689.73 620,944 489,303,872 28.071 9.237 789 2478.72 488927.99 622,521 491,169,069 28.089 9.240 790 2481.86 490168.14 624,100 493,039,000 28.107 9.244 791 2485.00 491409.85 625,681 494,913,671 28.125 9.248 792 2488.14 492653.14 627,264 496,793,088 28.142 9.252 793 2491.28 493898.20 628,849 498,677,257 28.160 9.256 .794 2494.43 495144.43 630,436 500,566,184 28.178 9.260 795 2497.57 496392.43 632,025 502,459,875 28.196 9.264 796 2500.71 497648.40 633,616 504,358,336 28.213 9.268 797 2503.85 498893.14 635,209 506,261,573 28.231 9.271 798 2506.99 500145.86 636,804 508,169,592 28.249 9.275 799 2510.13 501400.14 638,401 510,082,399 28.266 9.279 800 2513.28 502656.00 640,000 512,000,000 28.284 9.283 801 2516.42 503913.42 641,601 513,922,401 28.302 9.287 802 2519.56 505172.43 643,204 515,849,608 28.319 9.291 803 2522.70 506432.98 644,809 517,781,627 28.337 9.295 804 2525.84 507655.52 646,416 519,718,464 28.355 9.299 805 2528.98 508958.83 648,025 521,660,125 28.372 9.302 806 2532.12 510224.II 649,636 523,606,616 28.390 9.306 807 2535.27 511490.96 651,249 525,557,943 28.408 9.310 808 2538.41 512759.38 652,864 527,514,112 28.425 9.314 809 2541.55 514029.37 654,481 529,474,129 28.443 9.318 810 2544.09 515300.94 656,100 531,441,000 28.460 9.321 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 83 Number, or Crcum- Circular Square. Cube. Square Cube Dianwtw. ference. Area. Root. Root. i 811 2547.83 516574.07 657,721 533,411,731 28.478 9-325 812 2550-97 517848.77 659,344 535,387,328 28.496 9-329 813 2554-12 519125.05 660,969 537,366,797 28.513 9-333 814 2557.26 520402.85 662,596 539,353,144 28.531 9-337 815 2560.40 521682.31 664,225 541,343,375 28.548 9-341 S16 2563.54 522663.30 665,856 543,338,496 28.566 9-345 817 2566.68 524245.86 667,489 545,338,513 28.583 9-348 818 2569.82 525529.98 669,124 547,343,432 28.601 9.352 819 2572.97 526815,68 670,761 549,353,259 28.618 9-356 820 2576.11 528102.96 672,400 551,368,000 28.636 9.360 821 2579.25 529391.80 674,041 553,387,6^1 28.653 9-364 822 2582.39 530682.21 675,684 555,412,248 28.670 9-367 823 2585.53 531974.39 677,329 557,441,767 28.688 9.371 824 2588.64 533267.75 678,976 559,476,224 28.705 9-375 825 2591.82 534562.87 680,625 561,515,625 28.723 9.379 826 1 2594.96 535859.57 682,276 563,559,976 28.740 9-383 8a7 2598.10 537159-83 683,929 565,609,283 28.758 9-386 828 2601.24 538457-62 685,584 567,663,552 28.775 9.390 829 2604-38 539759.08 687,241 569,722,789 28.792 9-394 830 2607.52 541062.06 688,900 571,787,000 28.810 9.398 831 2610.66 542366.60 690,561 573.856,191 28.827 9.401 83* 2613.81 543672.72 692,224 575,930,368 28.844 9.405 833 2616.95 544980.52 693,889 578,009,537 28.862 9.409 834 2620.09 546289.68 695,556 580,093,704 28.879 9.413 i 835 2623.23 547600.51 697,225 582,182,875 28.896 9.417 836 2626.37 548912.91 698,896 584,277,056 28.914 9.420 837 2629.51 550226.89 700,569 586,376,253 28.931 9.424 838 2632.64 551542.43 702,244 588,480,472 28.948 9.428 839 263S.80 552859-58 703,921 590,589,719 28.965 9-432 840 2638.94 554178.24 705,600 592,704,000 28.983 9-435 84, 2642.08 555498.49 707,281 594,823,321 29.000 9-439 , 842 2645.22 556820.32 708,964 596,947,688 29.017 9-443 ! 843 2648.36 558143.72 710,649 599,077,107 29.034 9-447 844 2651.51 559468.69 712,336 601,211,584 29.052 9-450 84s 2654.65 560795.23 714,025 603,351,125 29.069 9-454 846 2657.79 562123.34 715,716 605,495,736 29.086 9.458 , 847 2660.93 563456.82 717,409 607,645,423 29.103 9.461 1 848 2664,07 564784.28 719,104 609,800,192 29.120 9.465 849 2667.21 566117.IO 720,801 611,960,049 29.138 9-469 , 850 2670.36 567451-59 722,500 614,125,000 29.155 9-473 1 851 2673.50 568787.46 724,201 616,295,051 29.172 9-476 852 ' 2676.64 570125.00 725,904 618,470,208 29.189 9.480 ' 853 2679.78 571464.10 727,609 620,650,477 29.206 9-483 ' 854 2682.92 572804.78 729,316 622,835,864 29.223 9.487 85s 2686.06 574147.03 731,025 625,026,375 29.240 9.491 ' 856 2689.20 575490.85 732,736 627,222,016 29-257 9-495 ' ^57 2692.35 576836.24 734,449 629,422,793 29-274 9-499 i 858 2695.49 578183.20 736,164 631,628,712 29.292 9.502 84 MATHEMATICAL TABLES. Number, or Circum- Circular Square. Cube. Square Cube Diameter. ference. Area. Root. Root. 859 2698.63 579531.73 737,881 633,839,779 29.309 9.506 860 2701.77 580881.84 739,600 636,056,000 29.326 9-509 861 2704.91 582233.51 741,321 638,277,381 29.343 9.513 862 2708.05 583586.75 743,044 640,503,928 29.360 9-517 863 2711.20 584941.57 744,769 642,735,647 29.377 9.520 864 2714-34 586297.95 746,496 644,972,544 29394 9-524 865 2717.48 587655-91 748,225 647,214,625 29.411 9.528 866 2720.66 589015.41 749,956 649,461,896 29.428 9532 867 2723.76 590376.54 751,689 651,714,363 29-445 9-535 868 2726.90 591739.20 753,424 653,972,032 29.462 9.539 869 2730.05 5<>3 103-44 755,161 656,234,909 29.479 9-543 870 2733-19 594469.26 756,900 658,503,000 29.496 9-546 871 2736.33 595836.44 758,641 660,776,311 29.513 9-550 872 2739.87 597205.59 760,384 663,054,848 29.529 9-554 873 2742.61 598576.91 762,129 665,338,617 29.546 9-557 874 2745-75 599948.21 763,876 667,627,624 29.563 9.561 875 2748.90 601321.87 765,625 669,921,875 29.580 9-565 876 2752.04 602697,11 767,376 672,221,376 29-597 9.568 877 2755-18 604073.91 769,129 674,526,133 29.614 9-572 878 2758.32 605451.49 770,884 676,836,152 29.631 9.575 879 2761.46 606832.24 772,641 679,151,439 29.648 9.579 880 2764.60 608213.76 774,400 681,472,000 29.665 9.583 881 2767.74 609596.84 776,161 683,797,841 29.682 9.586 882 2770.89 610981.50 777,924 686,128,968 29.698 9-590 883 2774.03 612367.74 779,689 688,465,387 29-715 9.594 884 2777.17 613755-54 781,456 690,807,104 29.732 9-597 885 2780.31 615144.91 783,225 693,154,125 29-749 9.601 886 2783-45 616535-85 784,996 695,506,456 29.766 9.604 887 2786.59 617928.37 786,769 697,864,103 29.782 9.608 888 2789-75 619322.45 788,544 700,227,072 29.799 9.612 889 2792.88 620718.11 790,321 702,595,369 29.816 9-615 890 2796.02 622115,34 792,100 704,969,000 29.833 9.619 891 2799.16 623514.13 793,881 707,347,971 29.850 9.623 892 2802.30 624914.50 795,664 709,732,288 29.866 9.626 893 2805.44 626316.44 797,449 712,121,957 29.883 9.630 894 2808.59 627719-95 799,236 714,516,984 29.900 9.633 895 2811.73 629120.35 801,025 716,917,375 29.916 9-637 896 2814.87 630531.68 802,816 719,323,136 29933 9.640 897 2818.82 631939-90 804,609 721,734,273 29.950 9.644 898 2821.15 633349-70 806,404 724,150,792 29.967 9.648 899 2824.29 634768.13 808,201 726,572,699 29.983 9-651 900 2827.44 636174.00 810,000 729,000,000 30.000 9.655 901 2830.58 637588.50 811,804 731,432,701 30.017 9.658 902 2833.72 639004.58 813,604 733,870,808 30.033 9.662 903 2836.86 640422.22 815,409 736,314,327 30.050 9.666 904 2840.00 641841.44 817,216 738,763,264 30.066 9.669 905 2843.14 643262,23 819,025 741,217,625 30.083 9.673 906 2846.28 644684.74 820,836 743,677,416 30.100 9.676 NUMBERS, OR DIAMETERS OF CIRCLES, &c. 85 Xumber, or Grcum- ference. 907 2849.43 908 2852.57 909 2855-71 910 2858.85 911 2861.99 912 2865.13 913 2868.29 914 2871.42 915 2874.56 916 2877.70 917 2880.84 918 2883.98 919 2887.13 920 2890.27 921 2893.41 922 2896.55 923 2899.69 924 2902.83 925 2905.98 926 2909.12 927 2912.26 928 2915.40 929 2918.54 930 2921.68 931 2924.82 932 2927.97 933 2931. II 934 2934.25 935 2937.39 936 2940.53 937 2943.67 938 2946.82 939 2949.96 940 2953.10 941 2956.24 942 2959.38 943 2962.43 944 2965.67 945 2968.81 946 2971.95 947 2975.09 948 2978.23 949 2981.37 950 2984.52 951 2987.66 952 2990.72 953 299394 954 2997.08 Ctrcttlar Area. Square. 646108.52 822,649 64753402 824,464 648961.09 826,281 650389.74 828,100 651819.95 829,921 653251-73 831,744 654689.09 833,569 656120.81 835,396 657556.51 837,225 658994.58 839,056 660432.22 840,889 661875.42 842,724 663318.20 844,561 664762.56 846,400 666208.48 848,241 667655-97 850,084 669IOI.61 851,929 670555-67 853,776 672007.87 855,625 673461.65 857,476 674916.99 859,329 676373-91 861,184 677832.40 863,041 679292.46 864,900 680754.08 866,761 682217.30 868,624 683682.06 870,489 685148.40 872,356 686616.31 874,225 688085.79 876,096 689556.85 877,969 691029.47 879,844 692503.67 881,721 693979-44 883,600 695456.77 885,481 696935-68 887,364 698416.14 889,249 699898.21 891,136 701381.83 893,025 702867.02 894,916 704350.25 896,809 705841.80 898,704 707332.02 900,601 708023.50 902,500 710316.54 904,401 7I181I.16 906,304 713307-34 908,209 714805.10 910,116 Cube. 746,142,643 748,613,312 751,089,429 753,571,000 756,058,031 758,550,528 761,048,497 763,551,944 766,060,875 768,575,296 771,095,213 773,620,632 776,151,559 778,688,000 781,229,961 783,777,448 786,330,467 788,889,024 791,453,125 794,022,776 796,597,983 799,178,752 801,765,089 804,357,000 806,954,491 809,557,568 812,166,237 814,780,504 817,400,375 820,025,856 822,656,953 825,293,672 827,936,019 830,584,000 833,237,621 835,896,888 838,561,807 841,232,384 843,908,625 846,590,536 849,278,123 851,971,392 854,670,349 857,375,000 860,085,351 862,801,408 865,523,177 868,250,664 Square Cube Root. Root. 30.116 9.680 30.133 9.683 30.150 9.687 30.163 9.690 30.183 9-694 30.199 9.698 30.216 9.701 30.232 9-705 30.249 9.708 30.265 9.712 30.282 9.715 30.298 9.718 30.315 9.722 30.331 9.726 30.348 9.729 30.364 9.733 30.381 9-736 30.397 9.740 30.414 9-743 30.430 9-747 . 30.447 9-750 30.463 9-754 30.479 9-757 30.496 9.761 30.512 9.764 30.529 9.768 30.545 9.771 30.561 9.775 30.578 9.778 30.594 9.783 30.610 9.785 30.627 9.789 30.643 9.792 30.659 9.796 30.676 9.799 30.692 9.803 30.708 9.806 30.724 9.810 30.741 9.813 30.757 9.817 30.773 9.820 30.790 9.823 30.806 9.827 30.822 9.830 30.838 9.834 30.854 9.837 30.871 9.841 30.887 9.844 86 MATHEMATICAL TABLES. j Number, or Circum- Circular Sauaie. Cube. Square Cube Diameter. ference. Area. »i^*i **•*• *p* Root. Root. 955 3000.22 716304.43 912,025 870,983,875 30.903 9.848 956 3003-36 717805.33 913,936 873,722,816 30.919 9.851 957 3006.51 719307.80 915,849 876,467,493 30.935 9.854 958 3009.65 720811.84 917,764 879,217,912 30.951 9.858 959 3012.79 722317.45 919,681 881,974,079 30.968 9.861 960 3015-93 723824.64 921,600 884,736,000 30.984 9.865 961 3019.07 725333-39 923»52i 887,503,681 31.000 9.868 962 3022.21 726843.71 925,444 890,277,128 31.016 9.872 963 3025.36 728355-61 927,369 893,056,347 31.032 9-875 964 3028.50 729869.07 929,296 895,841,344 31.048 9.878 965 3031.64 73 1384. 1 1 931,225 898,632,125 31.064 9.881 966 3034.78 732900.72 933,156 901,428,696 31.080 9.885 967 3037.92 734418.90 935,089 904,231,063 31.097 9.889 968 3041.06 735938.64 937,024 907,039,232 3^'^^3 9.892 969 3044.21 737459.96 938,961 909,853,209 31.129 9.895 970 3047.35 738982.86 940,900 912,673,000 31.145 9.899 971 3050.49 740507.32 942,841 915,498,611 31.161 9.902 972 3053.63 742033.35 944,784 918,330,048 31.177 9.906 973 3056.77 743560.95 946,729 921,167,317 31.193 9-909 • 974 3059.91 745090.13 948,676 924,010,424 31.209 9.912 975 3063.06 746620.87 950,625 926,859,375 31.225 9.916 976 3066.20 748153.19 952,576 929,714,176 31.241 9.919 977 3069.36 749687.07 954,529 932,574,833 31-257 9923 978 3072.48 751222.53 956,484 935,441,352 31.273 9.926 979 3075.62 752759-56 958,441 938,313,739 31.289 9.929 980 3078.76 754298.16 960,400 941,192,000 31-305 9-933 981 3081.90 755838.32 962,361 944,076,141 31-321 9-936 982 3085.05 757380.06 964,324 946,966,168 31.337 9.940 983 3088.19 758923.38 966,289 949,862,087 31.353 9-943 984 3091.33 760468.26 968,256 952,763,904 31.369 9.946 985 3094.47 762014.71 970,225 955,671,625 31-385 9.950 986 3097-61 763562.73 972,196 958,585,256 31.401 9-953 987 3100.75 765119-33 974,169 961,504,803 31.416 9-956 988 3103.96 766663.49 976,144 964,430,272 31.432 9.960 989 3107.04 768216.23 978,121 967,361,669 31.448 9963 990 3IIO.18 769770.54 980,100 970,299,000 31.464 9.966 991 3113.32 771326.41 982,081 973,242,271 31.480 9.970 992 3116.46 772883.86 984,064 976,191,488 31.496 9.973 993 3119.60 774442.88 986,049 979,146,657 31.512 9.977 994 3122.75 776003.47 988,036 982,107,784 31.528 9.980 995 3125.89 777565.63 990,025 985,074,875 31.544 9.983 996 3129.03 779129.36 992,016 988,047,936 31.559 9.987 997 3132.17 780694.66 994,009 991,026,973 31.575 9.990 998 3135-11 782261.54 996,004 994,011,992 31-591 9-993 999 3138.45 783829.98 998,001 997,002,999 31.607 9.997 1000 3141.60 785400.00 1,000,000 1,000,000,000 31.623 10.000 CIRCLES: — DIAMETER, CIRCUMFERENCE, &C. 87 TABLE No. IV. CIRCLES:— DIAMETER, CIRCUMFERENCE, AREA, AND SIDE OF EQUAL SQUARE. Diamecer. Grcum- fereoce. Area. Side of Equal Souare (Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). S/«6 H 1 t 9/16 ) H 'J" .1963 .3927 -5890 .7854 .9817 I.I781 . 1.3744 1.5708 1.7771 1.9635 2.1598 2.3562 2.5525 2.7489 2.9452 .00307 .01227 .02761 .04909 .07670 .1104 .1503 .1963 .2485 .3068 .3712 .4417 .5185 .6013 .6903 .0553 .1107 .1661 .2215 .2770 .3323 .3877 .4431 .4984 .5539 .6092 .6646 .7200 .7754 .8308 3 , 3'A 3 3/.« 3X 'J' 37/16 3 9A6 3H It 3'5/i6 9.4248 9.62 II 9.8175 10.014 10.210 10.406 10.602 10.799 10.995 II. 191 11.388 11.584 II.781 11.977 12.173 12.369 7.0686 7.3662 7.6699 7.9798 8.2957 8.6180 8.9462 9.2807 9.6211 9.9680 ia320 10.679 11.044 II.4I6 11.793 12,177 2.6586 2.7140 2.7694 2.8248 2.8801 2.9355 2.9909 3.0463 3.1017 3.1571 3.2124 3-2678 3-3232 3.3786 3.4340 3.4894 I I 5/:6 i 9/16 I"Vr6 3.1416 3.3379 3.5343 3-7306 3.9270 4.1233 4.3197 4.5160 4-7124 4-9087 5. 105 1 5.3014 5-4978 S.694I t.8905 60868 .7854 .8866 .9940 1.1075 1.227 1 1.3530 1.4848 1.6229 1.767 1 1.9175 2.0739 2.2365 2.4052 2.5800 2.761 1 2.9483 .8862 .9416 .9969 1.0524 I.IOI7 I.163I I.2185 1.2739 1.3293 1.3847 1. 4401 1.4955 1.5508 1.6062 I.6616 I.7170 43/16 4X 4 7/,6 49/16 4'S/i6 12.566 12.762 12.959 13-155 13-351 13-547 13-744 13.940 14.137 14.333 14.529 14.725 14.922 15.119 15.315 15.511 12.566 12.962 13-364 13-772 14186 14606 15.033 15-465 15.904 16.349 16.800 17.257 17.720 18.190 18.665 19.147 3.5448 3.6002 3.6555 3-7109 3.7663 3.8217 3.8771 3-9325 3.9880 4.0434 4.0987 4.1541 4.2095 42648 43202 4-3756 2 2 V16 2 3/,6 2 5/,6 ! 2^ 2 7/.6 2>^ 2 9/x6 2;i 2»»yi6 2'3/,6 2« 2'5/«6 1 6.2832 6^795 6.6759 6.J722 7.0686 7-2*49 7-4*13 7.6576 7.8540 8.0533 8.2^67 8.4430 8.6314 8.83f7 9.0311 3.1416 3.3380 3.5465 3.7584 3.9760 4.2000 4.4302 3.7066 4.9087 5-1573 5-4119 5.6723 5-9395 6.2126 6.4918 6.7772 1.7724 1.8278 1.8831 1.9385 1.9939 2.0493 2.1047 2. 1 601 2.2155 2.2709 2.3262 2.3816 2.4370 2.4924 2.5478 2.6032 5 5 '/16 5 3/,6 59/16 St 5'5/i6 15.708 15.904 16.100 16.296 16493 16.689 16.886 17.082 17.278 17.474 17-671 17.867 18.064 18.261 18.457 18.653 19.635 20.129 20.629 21.135 21.647 22.166 22.690 23.221 23.758 24301 24850 25.406 25.967 26.535 27.108 27.688 4.4310 44864 4-5417 4-5971 46525 4.7079 4.7633 48187 4.8741 4-9295 49848 5.0402 5-0956 5.1510 5.2064 5.2618 MATHEMATICAL TABLES. CIRCLES: — DIAMETER, CIRCUMFERENCE, &C. 89 Side of Side of .Diameter. Circum- ference. Area. Equal Sauare (Square Root of Area). Diameter. Circum- ference. Area. Equal Sauare (Square Root of Area}. iS 56.548 254.469 15-951 24 75-398 452.390 21.268 liyi 56.941 258.016 16.062 24}i 75-791 457-115 21.379 i&% 57.334 261.587 16.173 24X 76.183 461.864 21 490 tSH 57.726 265.182 16.283 24^ 76.576 466.638 21.601 18^ 58.119 268.803 16.394 24K 76.969 471.436 21.712 18H 58.512 272.447 16.505 24H 77.361 476.259 21.822 ii^ 58.905 276.117 16.616 24«' 77-7SA- 481.106 21.933 i»H 59.297 279.811 16.727 24;^ 78.147 485.978 22.044 19 59.690 283.529 16.837 25 78.540 490.875 22.155 19X 60.083 287.272 16.948 25 >i 78.932 495-796 22.265 19V 60.475 291.039 17.060 25 X 79-325 500.741 22.376 t9H 6a868 294.831 17.170 25^ 79718 505.711 22.487 •9>i 61.261 298.648 17.280 25 K 80.110 510.706 22.598 19^ 61.653 302.489 17.391 2SH 80.503 515-725 22.709 19^ 62.046 306.355 17.502 2sH 80.896 520.769 22.819 i9;< 62.439 310.245 17.613 25^ 81.288 525.837 22.930 1 - 20 62.832 314.160 17.724 26 81.681 530.930 23.041 20X 63.224 318.099 17.834 26% 82.074 536.047 23.152 20X 63.617 322.063 17.945 26X 82.467 541.189 23.062 20^ 64.010 326.05 I 18.056 26^ 82.859 546.356 23.373 70}4 64.402 330.064 18.167 26X 83.252 551-547 23484 20>i 64-795 334-101 18.277 26^ 83.645 556.762 23.595 204i^ 65.188 338.163 18.388 26X 84.037 562.002 23.708 20;< 65.580 342.250 18.499 26^ 84.430 567.267 23.816 21 65.973 346.361 18.610 27 84-823 572.556 23.927 21^ 66.366 350.497 18.721 27% 85.215 577.870 24.038 21X 66.759 354.657 18.831 27H 85.608 583.208 24.149 21H 67.151 358.841 18.942 27H 80.001 588.571 24.259 21X 67.544 363.05 1 19053 27 K 86.394 593.958 24.370 21^ 67.937 367.284 19.164 27H 86.786 599370 24481 . 21^ 68.329 371.543 19.274 27H 87.179 604.807 24.592 . 21^ 68.722 375.826 19.385 27% 87.572 610.268 24.703 1 22 69.115 380.133 19.496 28 87.964 615753 24.813 22X 69.507 384-465 19.607 28^ 88.357 621.263 24924 22 V 69.900 388.822 19.718 28X 88.750 626.798 25.035 22^ 70.293 393.203 19.828 28^ 89.142 632.357 25.146 5 22>i 70.686 397.608 19.939 28X 89.535 637.941 25.256 22K ji.orj^ 402.038 20.050 28;^ 89.928 643.594 25.367 -i^H 7^-471 406.493 20.161 28^ 90.321 649.182 25.478 22 J< 71.864 410.972 20.271 28?^ 90.713 654-839 25.589 23 72.256 415.476 20.382 29 91.106 660.521 25.699 23^ 72.649 420.004 20.493 29^ 91.499 666.227 25.810 23X 73.042 424-557 20.604 29X 91.891 671.958 25.921 23>i 73-434 429.135 20.715 29^ 92.284 677.714 26.032 , 23;^ 73.827 433.731 20.825 2rA 92.677 683.494 26.143 23H 74.220 438.363 20.936 29H 93.069 689298 26.253 . 23JK' 74.613 443.014 21.047 293^ 93-462 695.128 26.364 23^ 75.005 447.699 21.158 29;^ 93855 700.981 26.478 rX MATHEMATICAL TABLES. Sid= or Side of Dinmrtcr, ^— a™. Equal Smian iSquirrlt™. . a™. iSquilTRool Of Ana). 30 94.248 706.860 26.586 ^^. 113.097 10.7.87 31-903 30 Ji 94.640 712.762 26.696 36 J^ 113.490 1024.9s 32.014 30^ 95-033 718.690 26.807 36X 113-883 1032.06 32.124 30« 95.426 724.641 26.918 36« 114.27; 1039-19 32-235 3o;i 95-818 730.618 27.029 36X 1 14.668 1046.3 s 32-349 3o« 96.211 736.619 27.139 i(>H 115.061 ■053-52 1060.73 32.457 742.644 27.250 3^X 115.453 32-567 748.694 27.361 36^ MS.846 1067.95 32.678 7'; 4.760 27.472 37 1 16.239 1075.21 32.789 760.868 27.583 37 fi 1 [6,631 1082.48 32,900 766.992 27.693 37X 117,024 1089.79 33-011 773.140 27,804 37H 117.417 1097.11 33.021 779'3i3 27.915 y}% 117,810 1104.46 33.232 78;.sio 28.026 37H 1U..84 33-343 791.732 28. 136 yjH [•.l$l 1 1 [9.24 33.454 797.978 28.247 37Pi 1126.66 33564 804.249 28.358 3^ . 119,380 1.34.11 33.675 810.545 28.469 38 H 119-773 1.41.59 33786 816.865 28.580 38y 120.166 33.897 823.209 28.69. 3m 120,558 34.008 829.578 28.801 38 Ji 120,951 34-118 835-972 28.9.2 iSH 121.344 1171-73 34-229 842.390 29.023 3SH 121.737 1179.32 34.340 848.833 29.133 3i^ 122.1Z9 1.86.94 34-45" 855.30 29.244 39 122,522 1194-59 34.561 861.79 ^9.355 39>i 122,9.5 1202.26 34.672 868.30 29,466 39X 123.307 1209.95 34.783 874.84 ^9-577 39^ 123.700 1217.67 34.894 881.41 29,687 39^ 124.093 124-485 1225.42 35-005 J 888.00 29,798 39H 1233-18 35-115 894.61 29.909 39>^ .24.878 ! 240.98 35-226 ' 901.25 30.020 39H 125.271 1248,79 35-337 1 007.92 3o->3' 40 125.664 .256.64 35-448 914.61 30,241 40H 126.056 1264.50 3S-SSS 921.32 30.352 40V 126.449 1272.39 35.669 928.06 30.463 ^oH 126.842 .280.31 35780 934-82 30574 Ao'A 127.334 1288.25 35-891 941.60 30.684 AoH 127.627 1296.21 36.002 948.41 30.79s 40K 128.020 1304.20 36.112 955-25 30,906 4o?i 128.412 13.2.2I 36,223 962.11 31.017 41 128.805 1320.25 36.334. 968.99 31.128 4i>i 129.198 1328.32 36.445 975.90 31-238 41X 129.591 129.983 1336.40 36.555 982.84 31.349 41?^ 1344-5' 36.666 989.80 31.460 41^ 130.376 1352.65 36.777 996.78 31-571 4'X 130.769 1360.81 36,888 1003,78 31.681 41 Ji" 131. .61 1369.00 36.999 1010.82 31-792 At 'A 13'-SS4 1377-21 37-109 circles: — DIAMETER, CIRCUMFERENCE, &C. 91 DniBeter. Grcum- ference. Area. Side of Equal Sauare (Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Square (Square Root of Area). 1 42 42'A 42H 4J^ 131-947 132.339 132.732 133.125 133.518 i33.9'o 134.303 134.696 1385.44 1393-70 1401.98 1410.29 14x8.62 1426.98 1435-36 1443-77 37.220 37.331 37.442 37.552 37.663 37.774 37.885 37.996 48 48 >^ 48X 48>i 48^ 1 50.796 151.189 151.582 151-974 152.367 152.760 153.153 153.545 1809.56 1818.99 1828.46 1837.93 1847.45 1856.99 1866.55 1876.13 42.537 42.648 42.759 42.870 42.980 43.091 43.202 43.313 •« ! ;43H i j43V ; 43H 43>< .43M |43V •43J< 135.088 135.481 , 135.874 1 136.266 136.659 137.052 137.445 137.837 1452.20 1460.65 1469.13 1477-63 1486.17 1494.72 1503.30 I5II.90 38.106 38.217 38.328 38.439 38.549 38.660 38.771 38.882 49 ^ 49V 49H 49>i A9H 49H 49?^ 153.938 154-331 154-723 155.116 155.509 155.901 156.294 1 56.687 1885.74 1895.37 1905.03 1914.70 1924.42 1934-15 1943.91 1953-69 43.423 43.534 43.645 43.756 43-867 43.977 44.088 44.199 '44 44J< UH uH , 138.230 138.623 139.015 139.408 139.801 140.193 140.586 . 140.979 1520.53 1529.18 1537.86 1546.55 1555.28 1564.03 1572.81 1581.61 38.993 39.103 39.214 39.325 39.436 39.546 39657 39.768 50 50V 1 57.080 157.865 158.650 1 59.436 1963.50 1983.18 2002.96 2022.84 44.310 44.531 44.753 44.974 160.221 161.007 161.792 162.577 2042.82 2062.90 2083.07 2103.35 45-196 45-417 45-639 45.861 4> 1 141.372 141.764 142.157 142.550 142.942 143.335 143.728 144.120 1590.43 1599.28 1608.15 1617.04 1625.97 1634.92 1643.89 1652.88 39.879 39.989 40.110 40.2 1 1 40.322 40.432 40.543 40.654 45H ■45X AiH 45K 45^ 45V 52 , 52X 163.363 164.148 164.934 165.719 2123.72 2144.19 2164.75 2185.42 46.082 46.304 46.525 46.747 53 , S3X S3>i S3J< 166.504 167.290 168.075 168.861 2206.18 2227.05 2248.01 2269.06 46.968 47.190 47.411 47.633 46 46H 46X 46X 46>i 46M 46,V 46J< 144.513 144-906 145.299 145.691 146.084 146.477 146.869 147.262 1661.90 1670.95 1680.01 1689.10 1698.23 1707.37 1716.54 1725.73 40.765 40.876 40.986 41.097 41.208 41.319 41.429 41.540 54 , S4X S4>^ 54JC 169.646 170.431 I7I.217 172.002 2290.22 2311.48 2332.83 2354.28 47.854 48.076 48.298 48.519 SS 55K S5^ 172.788 173-573 174.358 175-144 2375.83 2397-48 2419.22 2441.07 48.741 48.962 49.184 49-405 47 47H '47V , 47« ,47>i i47M : i47V ! ,47J< 1 »- ____ 147.655 148.047 148.440 148.833 149.226 149.618 1 50.01 1 150.404 1734-94 1744.18 1753-45 1762.73 1772.05 1781.39 1790.76 1800.14 41.651 41.762 41.873 41.983 42.094 42.205 42.316 42.427 56 s(>'A 175.929 176.715 177.500 178.285 2463.01 2485.05 2507.19 2529.42 49.627 49.848 • 50.070 50.291 92 MATHEMATICAL TABLES. Diameter. Circum- ference. Area. Side of Equal Square l( Square Root of Area;. Diameter. Circum- ference. Area. Side of Equal Souare (Square Root of Area). 57 ^ 57X 57>i 57^ 179.071 179.856 180.642 181.427 2551.76 2574.19 2596.72 2619.35 50.513 50.735 50.956 51.178 68 68X 68^ 68^ 213.628 214.414 215.199 215.985 3631.68 3658UH 3685.29 3712.24 60.261 60.483 60.704 60.926 58 58X 58^ 182.212 182.998 183.783 184.569 2642.08 2664.91 2687.83 2710.85 51.399 51.621 51.842 52.064 69 69X 69X 69^ 216.770 217.555 218.341 219.126 3739.28 3766.43 3793.67 3821.02 61.147 61.369 61.591 61.812 59 , 59X 59>i 59^ 185.354 186.139 186.925 187.710 2733.97 2757.19 2780.51 2803,92 52.285 52.507 52.729 52.950 70 70X 70X 219.912 220.697 221.482 222.268 3848.46 3875.99 3903.63 3931.36 62.034 62.255 62.477 62.698 60 60X 60X 6o|<' 188.496 189.281 190.066 190.852 2827.44 2851.05 2874.76 2898.56 53.172 53.393 53.615 53.836 71 71X 71^ 223.053 223.839 224.624 225.409 3959.20 3987.13 4015.16 4043.28 62.920 63.141 63.363 63.545 61 61X 6i>i 6i?< 191.637 192.423 193.208 193.993 2922.47 2946.47 2970.57 2994.77 54.048 54.279 54.501 54.723 72 72^ 72 >i 72X 226.195 226.980 227.766 228.551 1 4071.51 4099.83 4128.25 4156.77 63.806 64.028 64.249 64.471 62 62X 62 >i 62«' 194.779 195.564 196.350 197.135 3019.07 3043.47 3067.96 3092.56 54.944 55.166 55.387 55.609 73 73X 7Z)i 229.336 230.122 230.907 231.693 4185.39 42 14. II 4242.92 4271.83 64.692 64.914 65.135 65.357 63 63^ 63>i 63^ 197.920 198.706 199.491 200.277 3117.25 3142.04 3166.92 319I.9I 55.830 56.052 56.273 56.495 74 74V U% 74H 232.478 233.263 234.049 234.834 4300.85 4329.95 4359.16 4388.47 65.578 65.800 66.022 66.243 64 64X 64>i 64l<' 201.062 201.847 202.633 203.418 3216.99 3242.17 3267.46 3292.83 56.716 56.938 57.159 57.381 75 , 7SX 75>i 7S)i 235.620 236.405 237.190 237.976 4417.87 4447.37 4476.97 4506.67 66u^65 66.686 66.908 67.129 65 6SX 65 >4 6SJ< 204.204 204.989 205.774 206.560 3318.31 3343.88 3369.56 3395.33 57.603 57.824 58.046 58.267 76 763^ 76;^ 76m: 238.761 239.547 240.332 241. 117 4536.47 4566.36 4596.35 4626.44 67.351 1 67.572 1 67.794 68.016 66 66X 66;^ 66m: 207.345 208.131 208.916 209.701 3421.20 3447.16 3473.23 3499.39 58.489 58.710 58.932 59.154 77 77X 77^ 773i 241.903 242.688 243.474 244.259 4656.63 4686.92 4717.30 4747.79 68.237 68.459 68.680 68.902 67X 67X 67X 210.487 211.272 ' 212.058 212.843 1 3525.66 3552.01 3578.47 3605.03 59.375 59.597 59.818 60.040 78 78X 78X 7»H 245.044 245.830 246.615 247.401 4778.37 4809.05 4839.83 4870.70 69.123 69.345 69.566 69.788 CIRCLES: — DIAMETER, CIRCUMFERENCE, &C 93 Dnuneter. Circuia- fcrence. Area. Side of Equal Souare (Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Souare (Square Root of Area). 79.'^ 79«' 248.186 248.971 249.757 , 250.542 4901.68 4932.75 4963.92 4995.19 70.009 70.231 70.453 70.674 90 90X 90X 9°}i 282.744 283.529 284.314 285.099 6361.74 6399.12 6432.62 6468.16 79.758 79.980 80.201 80.423 80 80V 80V 251.328 252.113 252.898 253.683 5026.56 5058.00 5089.58 5121.22 70.896 7I.II8 71.339 71.561 91 91X 9i>i 9«v: 285.885 286.670 287.456 288.242 6503.89 653968 6573.56 6611.52 80.644 80.866 81.087 81.308 81 8iy 8i>i 81 V 254.469 255.254 256.040 256.825 5153.00 5184.84 5216.82 5248.84 71.782 72.004 72.225 72.447 92 92X 92H 289.027 289.812 290.598 291.383 6647.62 6683.80 6720.07 6756.40 81.530 81.752 81.973 82.195 82 83V 82>i 82V 1 257.611 258.396 259.182 , 259.967 5281.02 5313.28 5345-62 5378.04 72.668 72.890 73.111 73.333 93 ^ 93^ 93X 93V 292.168 292.953 293.739 294.524 6792.92 6829.48 6866.16 6882.92 82.416 82.638 82.859 83.081 ■83 J83V 83}^ 83V < 260.752 ' 261.537 262.323 263.108 5410.62 5443.24 5476.00 5508.84 73.554 73.776 73.997 74.219 94, 94V 94^ 94V 295.310 296.095 296.881 297.666 6939.79 6976.72 7013.81 7050.92 83.302 83.524 83.746 83.968 84 84,V 84^ 84V 1 263.894 264.679 265.465 266.250 5541.78 5574.80 5607.95 5641.16 74.440 74.662 74.884 75.106 95 , 95X 95 >^ 9SH 298.452 299.237 300.022 300.807 7088.23 7125.56 7163.04 7200.56 84189 84.411 84632 84854 85 8;V 85V 267.036 267.821 268.606 269.392 5674.51 5707.92 5741.47 5775.09 75.327 75.549 75.770 75.992 96 96V 96'A 96V 301.593 302.378 302.164 303.948 7238.24 7275.96 7313.84 735172 85.077 85.299 85.520 85.742 186 86V 86>i 86V 270.177 270.962 271.748 272.533 5808.81 5842.60 5876.55 5910.52 76.213 76.435 76.656 76.878 97 97V 97 >i 97^ 304.734 305.520 306.306 307.090 7389.80 7427.96 7474.20 7504.52 85.963 86.185 86.407 86.628 87 87V ' 87>i ! 87V ; 273.319 274-104 274-890 275.675 5944.69 5978.88 6013.21 6047.60 77.099 77.321 77.542 77.764 98 98X 98^ 307.876 308.662 309.446 310.232 7542.96 7581.48 7620.12 7658.80 86.850 87.072 87.293 87.515 88 88V 88 Ji 88V 276.460 277.245 278.031 278.816 6082.13 6116.72 6151.44 6186.20 77.985 78.207 78.428 78.650 99 , 99^ 99'A 99^ 3II.OI8 311.802 312.588 313.374 7697.68 7736.60 7775.64 7814.76 87.736 87.958 88.180 88.401 100 100^ 314.159 315730 7854.00 7932.72 88.623 89.066 59 89V 89V 279.602 280.387 281.173 281.958 6221.15 6256.12 6291.25 6326.44 78.871 79.093 79.315 79.537 lOI 317.301 318.872 8011.84 8091.36 89.509 89.952 94 MATHEMATICAL TABLES. Diameter. Circum- ference. Area. Side of Equal Square ^Square Root of Area). Diameter. Circum- ference. Area. Side of Equal Souare (Square Root of Area). 1 02 I02X 320.442 322.014 8171.28 8251.60 90.395 90.838 112 II2X 351.858 353.430 9852.03 9940.20 99.258 99.701 103 IO3X 323.584 325.154 8332.29 8413.40 91.282 91.725 113 II3>^ 355.000 356.570 10028.75 loi 17.68 100.144 100.587 104 io4}4 326.726 328.296 8494.87 8576.76 92.168 92.61 1 114 358,142 359.712 10207.03 10296.76 101.03 1 101.474 105 329.867 331.438 8659.01 8741.68 93.054 93.497 115 . "5>^ 361.283 362.854 10386.89 10477.40 IOI.917 102.360 106 333.009 334.580 8824,73 8908.20 93.940 94.383 116 116;^ 364.425 365.996 10568.32 10659.64 102.803 103.247 107 107 >^ 336.150 337.722 8992.02 9076.24 94.826 95.269 117 367.566 369.138 10751.32 10843.40 103.690 104.133 108 108K 339.292 340.862 9160.88 9245.92 96.156 118 ii8>^ , 370.708 372.278 10935.88 1 1028.76 104.576 105.019 109 109;^ 342.434 344.004 9331.32 9417.12 96.599 97.042 119 119^ 373.849 375.420 1 1 122.02 1 121 5.68 105.463 105.906 no 345.575 347.146 9503.32 9589.92 97.485 97.928 120 376.991 M 309.73 106.350 III MiX 348.717 350.288 9676.89 9764.28 98.371 98.815 LENGTHS OF CIRCULAR ARCS. 95 TABLE No. v.— LENGTHS OF CIRCULAR ARCS FROM l^ TO l8o^ GIVEN, THE DEGREES. (Radius = i.) Degrees. Length. ' Degrees. 1 Length. Degrees. Length. Degrees. Length. I .0174 40 .6981 79 1.3788 117 2.0420 2 .0349 41 .7156 80 1-3963 118 2.0595 3 -0524 42 .7330 119 2.0769 4 .0698 43 .7505 81 1-4137 5 .0873 44 .7679 82 I.4312 120 2.0944 6 .0147 45 .7854 83 1.4486 121 2.II18 7 .0222 46 .8028 84 1. 4661 122 2.1293 8 .0396 47 .8203 85 1.4835 123 2.1467 9 -0571 48 .8377 86 I.50IO 124 2.1642 49 .8552 87 1.5^84 125 2.I8I7 lO .1745 ^ • w %/ 88 1-5359 126 2.1991 II .1920 50 .8727 89 1-5533 127 2.2166 12 .2094 SI .8901 ^% 128 2.2304 13 .2269 5» .9076 90 1.5708 129 2.2515 14 .2443 S3 .9250 91 L5882 15 .2618 54 .9424 92 1.6057 130 2.2689 16 .2792 SS •9599 93 L6231 131 2.2864 17 .2967 S6 •9774 94 1.6406 132 2.3038 18 -314I 57 .9948 95 L6581 133 2.3213 19 'ZZ^^ S8 1.0123 96 1.6755 134 2.3387 ^ 59 1.0297 97 1.6930 135 2.3562 ao .3491 98 L7I04 136 2.3736 21 .3665 60 1.0472 99 1.7279 137 2.391 1 22 .3840 61 1.0646 138 2.4085 23 .4014 62 1. 0821 100 1.7453 139 2.4260 24 .4189 63 10995 lot 1. 2628 1 25 •4363 64 1.1170 102 L7802 140 2.4435 26 .4538 65 I-I345 103 1.7977 141 2.4609 27 .4712 66 1.1519 104 1.8151 142 2.4784 28 .4887 67 1. 1694 105 1.8326 143 2.4958 29 .5061 68 1. 1868 106 1.8500 144 2.5133 ^ 69 1.2043 107 1.8675 145 2.5307 30 .5236 108 L8849 C46 2.5482 31 .54IC) 70 1. 2217 109 1.9024 147 2.5656 32 .5585 71 1.2392 148 2.5831 33 .5759 72 1.2566 no 1.9199 149 2.6005 34 .5934 73 1-2741 III 1-9373 . 35 .6109 , 74 L2915 112 1.9548 150 2.6180 36 .6283 75 1.3090 113 1.9722 151 2.6354 37 .645S ! 76 1.3264 114 1.9897 152 2.6529 38 .66:^2 77 1.3439 115 2.0071 153 2.6703 1 39 .6807 78 1.3613 ii6 2.0246 154 2.6878 i:"^i. 96 MATHEMATICAL TABLES. Degrees. Length. Degrees. Length. Degrees. Length. Degrees. Length. 155 2.7053 161 2.8100 168 2.9321 174 3.0369 156 2.7227 162 2.8274 169 2.9496 175 30543 157 2.7402 163 2.8449 176 3.0718 158 2.7576 164 2.8623 170 2.9670 177 3.0892 159 2.7751 165 2.8798 171 2.9845 178 3.1067 166 2.8972 172 3.0020 179 3. 1 241 160 2.7925 167 2.9147 173 30194 180 3.I416 L LENGTHS OP CIRCULAR ARCS. 97 TABLE No. VI.— LENGTHS OF CIRCULAR ARCS, UP TO A SEMICIRCLE. GIVEN, THE HEIGHT. (Chord = I.) Height. Length. HdghL Length. Height. Length. Height. Length. .100 1.02645 .140 I.05147 .180 1.08428 .220 1. 1 2445 .101 1.02698 .141 1.05220 .181 I.08519 .221 I.I2556 -I02 1.02752 .142 1.05293 .182 I.0861I .222 1. 12663 .103 1.02806 .143 1.05367 .183 1.08704 .223 1.12774 .104 1.02860 .144 I.0544I .184 1.08797 .224 1. 12885 -105 1. 02914 .145 I.05516 .185 1.08890 .225 1. 12997 .106 1.02970 .146 I.0559I .186 1.08984 .226 1.13108 .107 1.03026 .147 1.05667 .187 1.09079 .227 I.I3219 .108 1.03082 .148 1.05743 .188 I.O9174 .22-8 1.13331 .109 1-03139 .149 I.05819 .189 1.09269 .229 1. 13444 .110 I.O3I96 .150 1.05896 .190 1.09365 .230 I.I3557 .III 1.03254 .151 1.05973 .191 I.O9461 .231 I.I3671 .112 I.O3312 .152 1.0605 1 .192 1.09557 .232 I.I3786 ."3 I.O3371 .153 1. 06130 .193 1.09654 .233 1-13903 .114 1.03430 .154 1.06209 .194 1.09752 •234 1. 14020 .1^5 1.03490 •155 1.06288 .195 1.09850 .235 I.I4136 .116 I.O3551 .156 1.06368 .196 1.09949 .236 I.I4247 .117 I.O361I .157 1.06449 .197 1. 10048 .237 1-14363 .118 1.03672 .158 1.06530 .198 I.IOI47 .238 1. 14480 .119 1.03734 .159 I.0661I .199 1. 10247 .239 r. 14597 .120 1.03797 .160 1.06693 .200 1. 10348 .240 1.14714 .121 1.03860 .161 1.06775 .201 1. 10447 .241 1-14831 .122 1.03923 .162 1.06858 .202 I.IO548 .242 1. 14949 .123 1.03987 .163 I.0694I .203 1. 10650 .243 1. 15067 .124 1.0405 1 .164 1.07025 .204 I.IO752 .244 1.15186 •125 I.04I16 .165 1. 07109 .205 I.I0855 .245 1.15308 .126 I.04181 .166 I.07194 .206 I.IO958 .246 1.15429 .127 1.04247 .167 1.07279 .207 I.II062 .247 1.15549 .128 1.043 13 .168 1.07365 .208 I.II165 .248 1.15670 .129 1.04380 .169 I.07451 .209 I.II269 .249 1.15791 .130 1.04447 .170 1.07537 .210 I.II374 .250 1.15912 -131 I.O4515 .171 1.07624 .211 I.II479 .251 1. 16033 .132 1.04584 .172 I.O771I .212 I.II584 .252 1.16157 '^33 1.04652 .173 1.07799 .213 I.I1692 •253 1. 16279 •134 1.04722 .174 1.07888 .214 I.II796 .254 1. 16402 -135 1,04792 .175 1.07977 .215 1. 1 1904 .255 1. 16526 .136 1.04862 .176 1.08066 .216 I.I20II .256 1. 16649 •^37 1.04932 .177 I.08156 .217 I.I2II8 .257 1.16774 .13S •^39 / 1.05003 .178 1.08246 .218 1. 12225 .258 1. 16899 105075 \ .179 1.08337 .219 I.I2334 .259 1. 17024 98 MATHEMATICAL TABLES. Height, Length. .260 .261 .262 .263 .264 .265 .266 .267 .268 .269 .270 .271 .272 .273 .274 .275 .276 .277 .278 .279 .280 .281 .282 .283 .284 .285 .286 .287 .288 .289 .290 .291 .292 •293 .294 •295 .296 .297 .298 .299 .300 .301 .302 •303 •304 •305 .306 .17150 .17275 .17401 .17527 .17655 .17784 .17912 .18040 .18162 .18294 .18428 •18557 .18688 .18819 .18969 .19682 .19214 .19345 •19477 .19610 .19743 .19887 .20011 .20146 .20282 .20419 .20558 .20696 .20828 .20967 .21202 .21239 .21381 .21520 .21658 .21794 .21926 .22061 .22203 .22347 .22495 .22635 .22776 .22918 .23061 .23205 •23349 Height Length. .307 .308 .309 .310 .311 .312 .313 .314 .315 .316 .317 .318 .319 .320 .321 .322 .323 .324 .325 .326 •327 .328 •329 .330 •332 '333 .334 .335 •336 .337 .338 .339 .340 •341 .342 .343 .344 •345 .346 •347 .348 .349 .350 .351 .352 .353 •23494 .23636 .27780 •23925 .24070 .24216 .24360 .24506 .24654 .24801 .24946 .25095 .25243 .25391 •25539 .25686 .25836 .25987 •26137 .26286 .26437 .26588 .26740 .26892 .27044 .27196 .27349 .27502 .27656 .27810 .27864 .28118 .28273 .28428 .28583 .28739 .28895 .29052 .29209 .29366 •29523 .29681 •29839 .29997 .30156 .30315 .30474 Height. •354 •355 •356 •357 •358 .359 .360 .361 .362 '3^3 .364 .365 .366 .367 .368 .369 .370 .371 .372 .373 .374 .375 .376 .377 .378 .379 .380 .381 .382 .383 .384 .385 .386 .387 .388 .389 .390 .391 .392 •393 •394 .395 .396 .397 .398 •399 .400 Length. •30634 •30794 •30954 .31115 .31276 •31437 •31599 .31761 •3^923 .32086 •32249 .32413 •32577 .32741 •32905 .33069 .33234 .33399 .33564 .33730 .33896 .34063 .34229 .34396 .34563 .34731 .34899 .35068 •35237 .35406 •35575 •35744 .35914 .36084 .36254 .36425 .36596 .36767 •36939 .37111 .37283 .37455 .37628 .37801 .37974 .38148 .38322 Height. .401 .402 .403 .404 .405 .406 .407 .408 .409 .410 .411 .412 .413 .414 .415 .416 .417 .418 .419 .420 .421 .422 •423 .424 .425 .426 .427 .428 .429 .430 .431 .432 .433 .434 .435 .436 •437 .438 •439 .440 .441 .442 .443 .444 •445 .446 .447 Length. [.38496 .38671 .38846 •39021 .39196 39372 •30548 •39724 .39900 .40077 .40254 .40432 .40610 .40788 .409^6 .41145 ■41324 ■41503 .41682 .41861 .42041 .42222 .42402 •42583 .42764 •42945 •43127 •43309 .43491 •43673 •43856 •44039 .44222 .44405 .44589 •44773 •44957 ■45142 ■45327 •455" •45697 ■45883 .46069 •46255 .46441 .46628 .46815 LENGTHS OF CIRCULAR ARCS. 99 Hdght. .448 •449 .450 .451 •452 •453 •454 •455 •45 <^ •457 .458 •459 JLength. 1.47002 1. 47189 1.47377 1.47565 1.47753 1.47942 I.48131 1.48320 1.48509 1.48699 1.48889 1.49079 .460 1.49269 Height. .461 .462 .463 .464 .465 .466 .467 .468 .469 .470 .471 .472 .473 .474 Length. .49460 .49651 .49842 •50033 .50224 .50416 .50608 .50800 .50992 •51185 •51378 ■5157I .51764 •51958 Height Length. .475 .476 .477 .478 .479 480 .481 .482 .483 .484 .485 .486 .487 .488 52^52 52346 52541 52736 52931 53126 53322 53518 53714 53910 54106 54302 54499 54696 Height .489 .490 .491 .492 •493 •494 •495 496 .497 .498 499 .500 Length. 1-54893 55090 .55288 .55486 .55685 •55854 .56083 .56282 .56481 .56680 .56879 -57079 MATHEMATICAL TABLES. TABLE No. VIL— AREAS OF CIRCULAR SEGMENTS, UP TO A SEMICIRCLE. AREAS OF CIRCULAR SEGMENTS. 101 , HdghL 1 1 Area. Height. Area. Height. Area. Height. 1 1 Area. '.X57 .07892 .203 .11423 .249 ; .15268 •295 .19360 .158 .07965 ' .204 .11504 .250 • IS3SS .296 . -19451 •159 .08038 .205 .11584 •251 .252 .15442 .15528 .297 •19543 .160 .08111 .206 .11665 .298 .19634 .161 .162 .08185 .08258 .207 .208 >k ^v ^^ .11746 .11827 .11908 .253 .254 .15615 .15702 .299 .300 •19725 .19817 .163 .08332 .209 .255 .15789 .301 .19908 .164 .08406 .210 .11990 .256 .15876 .302 .20000 .165 .08480 .211 I2071 .257 .15964 •303 .20092 . .166 .08554 .212 I2153 .258 .16051 .304 .20184 .167 .08629 .213 12235 •259 .16139 •305 .20276 .168 1 ^ .08704 .214 I2317 .260 .16226 .306 .20368 .169 .08778 .215 . .12399 .261 .16314 •307 .20460 .170 .08854 .216 1 248 1 .262 .16402 .308 •20553 .171 .08929 .217 12563 .263 .16490 •309 .20645 .172 .09004 .218 12646 .264 .16578 .310 .20738 .173 .09080 .219 12729 .265 16666 %0 •3" .20830 .174 •09155 .220 I281I .266 1675s .312 .20923 •175 .09231 .221 12894 .267 16843 •313 .21015 .176 .09307 .222 12977 .268 16932 •314 .21108 •'^2 -09383 .223 13060 .269 17020 •315 .21201 .178 .09460 .224 I3144 .270 .17109 .316 .21294 .179 .09537 .225 13227 .271 .17198 .317 .21387 .180 .09613 .226 133" .272 .17287 .318 .21480 .181 .09690 .227 13395 .273 .17376 •319 •21573 . .182 .09767 .228 13478 .274 .17465 .320 .21667 .183 .09845 .229 13562 .275 17554 •J •321 .21760 .184 .09922 .230 13646 .276 .17644 .322 •21853 i-'f5 .09200 .231 I373I .277 'I7733 •323 .21947 .186 .10077 .232 I3815 .278 .17823 .324 .22040 .187 €% .10153 .233 13899 .279 .17912 .325 .22134 .188 .10233 .234 13984 .280 .18002 .326 .22228 .189 .10317 .235 14069 .281 .18092 .327 .22322 .190 .10390 .236 I4I54 .282 .18182 .328 .22415 ,.191 .10469 .237 14239 .283 .18272 •329 .22509 .192 .10547 .238 14324 .284 .18362 •330 •332 •333 .334 •335 .22603 .22697 .22792 .22886 .22980 .23074 •193 .194 .195 .196 .197 .198 .» ^ _ .10626 .10705 .10784 .10864 .10943 •239 • .240 .241 .242 .243 14409 14494 14580 14665 14752 .285 .286 .287 .288 .289 .18452 .18542 .18633 .18723 .18814 .11023 .244 •14837 .290 .18905 .336 .23169 1 .199 .11102 .245 14923 .291 .18996 •337 .23263 .200 .11182 .246 .15009 .292 .19086 •338 •23358 .201 .11262 .247 .15096 •293 .19177 •339 •23453 .202 .11343 .248 .15182 •294 .19268 .340 .^3547 SINES, COSINES, &C OF ANGLES. 103 TABLE No. VIIL— SINES, COSINES, TANGENTS, COTANGENTS, SECANTS, AND COSECANTS OF ANGLES FROM 0° to 90°- Advancing by 10' or one-sixth of a degree. (Radius =i.) 1 Angle. Sine. Cosecant. Tangent Cotangent Secant • Cosine. 0' 0' .000000 Infinite. .000000 Infinite. 1. 00000 ] [.000000 90° 0' 10 .002909 343-77516 .002909 343.77371 1. 00000 .999996 50 20 .005818 171.88831 .005818 171.88540 1.00002 .999983 40 1 30 .008727 114.59301 .008727 114.58865 1.00004 .999962 30 40 .011635 85-945609 .011636 85.939791 1.00007 .999932 20 50 .014544 68.757360 .014545 68.750087 1. 000 1 1 .999894 10 I .017452 57.298688 .017455 57.289962 I.OOOI5 .999848 89 10 .020361 49.114062 .020365 49.103881 1.0002 1 .999793 50 20 .023269 42.975713 .023275 42.964077 1.00027 .999729 40 1 30 .026177 38.201550 .026186 38.188459 1.00034 •999657 30 40 .029085 34.382316 .029097 34.367771 1.00042 .999577 20 ' 50 .031992 31.257577 .032009 31.241577 1.0005 1 .999488 10 2 .034899 28.653708 .034921 28.636253 1.0006 1 .999391 88 ' 10 .037806 26.450510 •037834 26.431600 1.00072 .999285 50 20 .040713 24.562123 .040747 24.541758 1.00083 .999171 40 30 .043619 22.925586 .043661 22.903766 1.00095 .999048 30 40 .046525 21.493676 .046576 21.470401 1. 00108 .998917 20 SO .049431 20.230284 .049491 20.205553 1. 00122 .998778 10 3 P .052336 19.107323 .052408 19.081 137 I.OOI37 .998630 87 10 •055241 18.102619 .055325 18.074977 I.OOI53 .998473 50 20 .058145 17.198434 .058243 17-169337 1. 00169 .998308 40 30 .061049 16.380408 .061163 16.349855 1. 00187 •998135 30 40 .063952 15.636793 .064083 15.604784 1.00205 .997357 20 50 .066854 14.957882 .067004 14.924417 1.00224 •997763 10 4 .069756 14.335587 .069927 14.300666 1.00244 .997564 86 10 .072658 i3.763"5 .072851 13.726738 1.00265 .997357 50 20 •075559 13.234717 .075776 13.196888 1.00287 .997141 40 30 .078459 12.745495 .078702 12.706205 1.00309 .996917 30 40 .081359 12.291252 .081629 12.250505 1.00333 .996685 20 50 .084258 11.868370 .084558 II. 826167 1.00357 .996444 10 5 .087156 11-473713 .087489 11.430052 1.00382 .996195 85 10 .090053 11.104549 .090421 11.059431 1.00408 .995937 50 20 .092950 10.758488 .093354 10.711913 1.00435 .995671 40 30 .095846 10.433431 .096289 10.385397 1.00463 .995396 30 40 .098741 10.127522 .099226 10.078031 1.0049 1 •995113 20 50 .101635 9.8391227 .102164 9.7881732 1.005 2 1 .994822 10 Angle. Cosine. Secant Cotangent Tangent Cosecant Sme. I04 MATHEMATICAL TABLES. Angle. Sine. Cosecant. Tangent. Cotangent Secant Cosine. 6° o' .104528 9.5667722 .105104 95143645 1.0055 X •994522 84° 0' lO .107421 9.3091699 .X 08046 92553035 X. 00582 .9942x4 50 20 .110313 9.0651512 .X 10990 9.OO9826X X.O0614 •993897 40 30 .113203 8.8336715 .113936 8.7768874 1.00647 .993572 30 40 .116093 8.613790I .XX6883 8.5555468 i.oo68x •993238 20 50 .118982 8.4045586 •119833 8.3449558 1.007x5 .992896 10 7 .121869 8.2055090 .122785 8.x 443464 1.0075X .992546 83 10 .124756 8.0156450 .125738 7.9530224 X.00787 .992x87 50 20 .127642 7-8344335 .X 28694 7.7703506 1.00825 .99x820 40 30 .130526 7.6612976 .131653 7.5957541 X. 00863 •991445 30 40 •I334IO 7.4957100 .134613 7.4287064 X.00902 .99106X 20 50 .136292 7^3371909 .137576 7.2687255 X.00942 .990669 10 8 •I39173 7.1852965 .140541 7^ii53697 X. 00983 .990268 82 10 •142053 7.0396220 •143508 6.9682335 X. 0x024 .989859 50 20 .144932 6.8997942 .146478 6.8269437 X. 0x067 •989442 40 30 .147809 6.7654691 .X 4945 1 6.691x562 x.oxxxx .989016 30 40 .150686 6.6363293 .152426 6.5605538 I-OII55 .988582 20 50 •I53561 6^5120812 .155404 6.4348428 1.0x200 .988x39 10 9 .156434 6.3924532 .158384 6.3137515 1.01247 .987688 81 10 •159307 6.2771933 .161368 6.X970279 X. 01294 .987229 50 20 .162178 6.1660674 •164354 6.0844381 1.0x432 .986762 40 30 .165048 6.0588980 .167343 59757644 X.OI39I .986286 30 40 .167916 59553625 •170334 5.8708042 1.0x440 .98580X 20 50 .170783 5.8553921 .173329 5.7693688 X.OI49I •985309 XO 10 .173648 5-7587705 .176327 5.6712818 1-01543 .984808 80 10 .176512 5-6653331 .179328 5-5763786 1^01595 .984298 * 50 20 •179375 5-5749258 .182332 5.4845052 X. 0x649 .983781 40 30 .182236 5.4874043 •185339 5.3955172 X.OI703 •983255 30 40 .185095 5.4026333 .188359 5^3092793 1.0x758 .98272X 20 50 •187953 5.3204860 .191363 5.2256647 x.ox8x5 .982x78 10 II .190809 5.2408431 .194380 5.1445540 X. 0x872 .98x627 79 10 .193664 5.1635924 .X9740X 5-0658352 X. 01930 .981068 50 20 .196517 5.0886284 .200425 4.9894027 1.0x989 .980500 40 30 .199368 5.OI58317 .203452 49151570 1.02049 •979925 30 40 .202218 4.9451687 .206483 4.8430045 1.02110 •979341 20 50 .205065 4.8764907 .209518 4.7728568 1.0217X .978748 10 12 .207912 4.8097343 .212557 4.7046301 1.02234 .978148 78 10 .210756 4.7448206 •215599 4.6382457 1.02298 •977539 SO 20 •213599 4.6816748 .218645 4-5736287 1.02362 .976921 40 30 .216440 4.6202263 .22x695 4.5x07085 1.02428 .976296 30 40 .219279 4.5604080 .224748 4.4494x81 1.02494 .975662 20 50 .2221X6 4-502x565 .227806 4,3896940 1.02562 .975020 10 1 Coune. Secant Cotangent Tangent Cosecant Sine. Angle. SINES, COSINES, &C OF ANGLES. 105 Angle. Sine. Cosecant. Tangent. Cotangent Secant. Cosine. 13° 0' .224951 4.44541 15 .230868 4.3314759 1.02630 •974370 77" 0' 10 .227784 4.3901 158 •233934 4.274706*6 1.02700 .973712 50 20 .230616 4.3362150 .237004 4.2193318 1.02770 •973045 40 30 •233445 4-2836576 .240079 4.1652998 1.02842 .972370 30 40 .236273 4^2323943 .243158 4.II25614 1. 02914 .971687 20 50 .239098 4,1823785 .246241 4.0610700 1.02987 •970995 10 14 .241922 4.1335655 .249328 4,0107809 I.O3061 .970296 76 1 10 •244743 4.0859130 .252420 3.9616518 IO3137 .969588 50 ' 20 •247563 4.0393804 •255517 3.9136420 I.O3213 .968872 40 30 .250380 3.9939292 .258618 3.866713I 1.03290 .968148 30 40 •253195 3.9495224 .261723 3.8208281 1.03363 .967415 20 50 .256008 3.9061250 .264834 3.7759519 1.03447 .966675 10 15 .258819 3.8637033 •267949 3.7320508 1.03528 .965926 75 10 .261628 3.8222251 .271069 3.6890927 1.03609 .965169 50 20 .264434 3.7816596 .274195 3.6470467 I.O369I .964404 40 30 .267238 3^74i9775 •277325 3.6058835 1.03774 .963630 30 40 .^70040 3.7031506 .280460 3.5655749 1.03858 .962849 20 50 .272840 3.6651518 .283600 3.5260938 i.o39'44 .962059 10 16 •275637 3.6279553 .286745 3.4874144 1.04030 .961262 74 10 .278432 3-5915363 .289896 3.4495120 1.04117 .960456 50 20 .281225 3-5558710 .293052 3.4123626 1.04206 .959642 40 , 30 .284015 3.5209365 .296214 3.3759434 1.04295 .958820 30 40 .286803 3.48671 10 .299380 3.3402326 1.04385 •957990 20 50 .289589 3^4531735 •302553 3.3052091 1.04477 .957151 10 17 .292372 3.4203036 .305731 3.2708526 1.04569 •956305 73 10 •295152 3.3880820 .308914 3.2371438 1.04663 .955450 50 20 .297930 33564900 .312104 3.2040638 1.04757 •954588 40 30 .300706 3^3255095 .315299 3.1715948 1.04853 •953717 30 40 •303479 3.2951234 .318500 3-1397194 1.04950 •952838 20 50 .306249 3.2653149 .321707 3.I0842IO 1.05047 •951951 10 18 .309017 3.2360680 .324920 3-0776835 1.05146 •951057 72 10 .311782 3.2073673 .328139 3.0474915 1.05246 •950154 50 20 •314545 3.1791978 •331364 3.OI783OI 1-05347 .949243 40 30 •317305 3-1515453 .334595 2.9886850 1.05449 .948324 30 40 .320062 3.1243959 .337833 2.9600422 1.05552 .947397 20 50 .322816 30977363 •341077 2.9318885 1.05657 .946462 10 19 •325568 3^o7i5535 .344328 2.9042109 1.05762 •945519 71 10 •328317 3^0458352 .347585 2.8769970 1.05869 .944568 50 20 •331063 3.0205693 .350848 2.8502349 1.05976 •943609 40 30 •333807 2.9957443 .354119 2.8239129 1.06085 .942641 30 40 •336547 2.9713490 .357396 2.7980198 1.06195 .941666 20 50 •339285 2.9473724 .360680 2.7725448 1.06306 .940684 10 Cosine. Secant. Cotangent. Tangent Cosecant. i Sine. Angle. io6 MATHEMATICAL TABLES. Angle. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. 20** O' .342020 2.9238044 •363970 2.7474774 1. 06418 •939693 70° 0' lO •344752 2.9006346 .367268 2.7228076 I.0653I .938694 50 20 .347481 2.8778532 •370573 2.6985254 1.06645 .937687 40 30 .350207 2.8554510 •373885 2.6746215 1. 06761 .936672 30 40 •352931 2.8334185 •377204 2.6510867 1.06878 .935650 20 50 •355651 2.8117471 .380530 2.627912I 1.06995 •934619 10 21 .358368 2.7904281 .383864 2.6050891 I.07115 •933580 69 10 .361082 2.7694532 .387205 2.5826094 ^•07235 •932534 50 20 •363793 2.7488144 .390554 2.5604649 1.07356 .931480 40 30 .366501 2.7285038 .393911 2.5386479 1.07479 .930418 30 40 .369206 2.7085139 .397275 2.5171507 1.07602 .929348 20 50 .371908 2.6888374 .400647 2.4959661 1.07727 .928270 10 22 .374607 2.6694672 .404026 2.4750869 107853 .927184 68 10 ■377302 2.6503962 .407414 2.4545061 1. 07981 .926090 50 20 •379994 2.6316180 .410810 2.4342172 1. 08109 .924989 40 30 .382683 2.6131259 .414214 2.4142136 1.08239 .923880 30 40 •385369 2.5949137 .417626 2.3944889 1.08370 .922762 20 SO .388052 2.5769753 .421046 2.3750372 1.08503 .921638 10 23 .390731 2.5593047 .424475 2.3558524 1.08636 •920505 67 10 •393407 2.5418961 .427912 2.3369287 I.08771 .919364 50 20 .396080 2.5247440 •431358 2.3182606 1.08907 .918216 40 30 •398749 2.5078428 .434812 2.2998425 1.09044 .917060 30 40 .401415 2.491 1874 .438276 2.2816693 I.O9183 .915896 20 50 .404078 2.4747726 .441748 2.2637357 1.09323 .914725 10 24 .406737 2.4585933 •445229 2.2460368 1.09464 •913545 66 10 .409392 2.4426448 .448719 2.2285676 1.09606 •912358 50 20 .412045 2.4269222 .452218 2.2II3234 1.09750 .911164 40 30 .414693 2.4II42IO .455726 2.1942997 1.09895 .909961 30 40 .417338 2.3961367 .459244 2.1774920 I.IOO4I .908751 20 SO .419980 2.3810650 .462771 2.1608958 I.IO189 .907533 10 25 .422618 2.3662016 .466308 2.1445069 I.IO338 .906308 65 10 .425253 2.3515424 •469854 2.I283213 1. 10488 •905075 50 20 .427884 2.3370833 •473410 2. 1 1 23348 1. 10640 •903834 40 30 •4305 1 1 2.3228205 •476976 2.0965436 I.IO793 •902585 30 40 •433^35 2.3087501 .480551 2.0809438 1. 10947 .901329 20 so •435755 2.2948685 .484137 2.0655318 I.III03 .900065 10 26 •438371 2.281172O .487733 2.0503038 I.II260 .898794 64 10 .440984 2.2676571 .491339 2.0352565 I.II419 .897515 50 20 •443593 2.2543204 .494955 2.0203862 I.II579 .896229 40 30 .446198 2.24II585 .498582 2.0056897 1. 1 1 740 .894934 30 40 .448799 2.2281681 .502219 I.99II637 I.I 1903 .893633 20 SO •451397 2.2153460 .505867 1.9768050 1. 12067 .892323 10 • Cosine. Secant. Cotangent. Tangent. Cosecant Sine. Angle. SINES, COSINES, &C OF ANGLES. 107 Aflgk. Sine. Cosecant. Tangent. Cotangent. Secant. Cosine. 27*^0' •453990 2.2026893 •509525 1. 9626105 1. 12233 ' ,891007 63° 0' 10 .456580 2.I901947 •513195 1.9485772 1. 1 2400 .889682 50 20 .459166 2.1778595 .516876 1.9347020 I.I2568 .888350 40 30 .461749 2.1656806 .520567 1. 9209821 1. 12738 .887011 30 40 .464327 2.1536553 .524270 I.9074147 1. 12910 .885664 20 50 .466901 2.I417808 .527984 1.893997 1 1. 13083 .884309 10 28 .469472 2.1300545 .531709 1.8807265 I.I3257 .882948 62 10 .472038 2.II84737 •535547 1,8676003 I- 13433 .881578 50 20 .474600 2.1070359 •539195 I.8546159 1. 13610 .880201 40 30 .477159 2.0957385 .542956 I.8417409 I.I3789 .878817 30 40 •479713 2.0845792 .546728 1.8290628 1. 13970 .877425 20 50 .482263 2.0735556 •550515 1. 8164892 I.I4152 .876026 10 2g .4848x0 2.0626653 .554309 1.8040478 1-14335 .874620 61 10 .487352 2.0519061 .558118 1. 7917362 1.14521 .873206 50 20 .4S989O 2.0412757 •561939 1.7795524 1.14707 .871784 40 30 .492424 2.0307720 .565773 1.7674940 1. 14896 .870356 30 40 •494953 2.0203929 .569619 1.7555590 1.15085 .868920 20 50 •497479 2.OIOI362 .573478 '.7437453 1.15277 .867476 10 30 .500000 2.0000000 .577350 1.7320508 i.15470 .866025 60 10 •502517 T.9899822 .581235 1.7204736 1.T5665 .864567 50 20 .505030 T. 98008 10 •585134 I.709OT16 1.15861 .863102 40 30 •507538 1.9702944 .589045 I.697663I 1. 16059 .861629 30 40 .510043 1.9606206 .592970 1.6864261 1.16259 .860149 20 50 •512543 1.95^0577 .596908 1.6752988 1.16460 .858662 10 31 .515038 1. 9416040 .600861 1.6642795 1.16663 .857167 59 10 •517529 1.9322578 .604827 1.6533663 1.16868 .855665 50 20 .520016 I.923OI73 .608807 1.6425576 1.17075 .854156 40 30 .522499 1.9138809* .612801 1.6318517 1.17283 .852640 30 40 •524977 1.9048469 .616809 1.6212469 1.17493 .851117 20 50 .527450 I.8959138 .620832 1. 6107417 1.17704 .849586 10 32 .529919 1.8870799 .624869 1.6003345 1.17918 .848048 58 10 .532384 1.8783438 .628921 1.5900238 1.18133 .846503 50 20 .534844 1.8697040 .632988 1.5798079 1.18350 .844951 40 30 •537300 i.8611590 .637079 1.5696856 1.18569 .843391 30 40 •539751 1.8527073 .641167 1-5596552 1.18790 .841.825 20 50 .542197 1.8443476 .645280 '•5497155 1. 19012 .840251 . 10 33 .544639 1.8360785 .649408 1.5398650 1.19236 .838671 57 10 .547076 1.8278985 .653531 1. 5301025 1. 19463 -837083 50 20 .549509 I.8198065 .657710 1. 5204261 1. 19691 .835488 . 40 30 •551937 I.81180IO .661886 I.5108352 1.19920 .833886 30 40 .554360 1.8038809 .666077 I.5013282 1.20152 .832277 20 50 .556779 1.7960449 .670285 I.4919039 1.20386 .830661 10 OxBoe. Secant. Cotangent. Tangent Cosecant. 1 Sine. Angle. io8 MATHEMATICAL TABLES. Angle. Sine. Cosecant Tangent. Cotangent. Secant. Co^e. • 34° o' •559193 1.7882916 •674509 1. 4825610 1.20622 .829038 56° 0' lO .561602 1.7806201 •678749 1.4732983 1.20859 .827407 50 20 .564007 1.7730290 .683007 I.464II47 1. 21099 .825770 40 30 .566406 I.7655173 .687281 1.4550090 1.21341 .824126 30 40 .568801 1.7580837 .691573 I.445980I I.21584 .822475 20 50 •571191 1.7507273 .695881 1.4370268 1. 21830 .820817 10 35 •573576 1.7434468 .700208 1. 4281480 1.22077 .819152 55 10 •575957 1. 7362413 .704552 I.4193427 1.22327 .817480 50 20 •578332 1. 7291096 .708913 1. 4106098 1.22579 .815801 40 30 .580703 1.7220508 .713293 I.4OI9483 1.22833 .814116 30 40 .583069 I.7150639 .717691 1-3933571 1.23089 .812423 20 50 •5S5429 1. 7081478 .722108 1.3848355 1.23347 .810723 10 36 •587785 I.7013016 .726543 1.3763810 1.23607 .809017 54 10 .590136 1.6945244 .730996 ^•3679959 1.23869 .807304 50 20 .592482 I.687815I .735469 1.3596764 1. 24134 .805584 40 30 .594823 1. 681 1730 .739961 1.3514224 1.24400 .803857 30 40 •597159 1.6745970 .744472 1.3432331 1.24669 .802123 20 50 .599489 1.6680864 .749003 1.3351075 1.24940 .800383 10 37 .601815 I.661640I .753554 1.3270448 1. 25214 .798636 53 10 .604136 1-6552575 .758125 1.3190441 1.25489 .796882 50 20 .606451 1.6489376 .762716 1.3 1 1 1046 1.25767 .795121 40 30 .608761 1.6426796 .767627 1.3032254 1.26047 •793353 30 40 .611067 1.6364828 .771959 1.2954057 1.26330 .791579 20 50 .613367 1.6303462 .776612 1.2876447 1. 26615 .789798 10 38 .615661 1.6242692 .781286 1.2799416 1.26902 .788011 52 10 .617951 I.618251O .785981 1.2722957 I.2719I .786217 50 20 .620235 1. 6122908 .790698 1.2647062 1.27483 .784416 40 30 .622515 1.6063879 .795436 1.2571723 1.27778 .782608 30 40 .624789 1. 6005416 .800196 1.2496933 1.28075 .780794 20 50 .627057 I.594751I .804080 1.2422685 1.28374 .778973 10 39 .629320 I.589OI57 .809784 1.2348972 1.28676 .777146 51 10 •631578 1.5833318 .814612 1.2275786 1.28980 .775312 50 20 .633831 1-5777077 .819463 1.2203121 1.29287 .773472 40 30 .636078 1.5721337 .824336 1. 2130970 1.29597 .771625 30 40 .638320 I.566612I .829234 1.2059327 1.29909 •769771 20 SO .640557 I.5611424 .834155 1.1988184 1.30223 .767911 10 40 .642788 1.5557238 .839100 1.1917536 I.3054I .766044 50 10 .645013 1.5503558 .844069 1. 1847376 1. 30861 .764171 50 20 .647233 1.5450378 .849062 1.1777698 1.3^83 .762292 40 30 .649448 1.5397690 .854081 1. 1 708496 I.31509 .760406 30 40 •651657 1.5345491 .859124 1. 1639763 1.31837 .758514 20 50 .653861 1.5293773 .864193 1.1571495 I.32168 .756615 10 Cosine. Secant. Cotangent. Tangent. Cosecant. Sine. Angle. SINES, COSINES, &c. OF ANGLES. 109 Angle. 1 Sixie. Cosecant. Tangent. Cotangent. Secant. Cosine. « 41° 0' .656059 I.524253I .869287 1. 1503684 1.32501 •754710 49° 0' 10 .658252 I.519I759 .874407 1. 1436326 1.32838 .752798 50 20 .660439 I.514I452 .879553 I.I369414 I.33177 .750880 40 30 .662620 I.5091605 .884725 1. 1302944 I.33519 .748956 30 40 .664796 I.50422II .889924 1. 1236909 1.33864 .747025 20 50 .666966 1.4993267 .895151 I.II71305 1.34212 .745088 10 142 .669131 1.4944765 .900404 I.II06125 1.34563 .743145 48 ID .671289 1.4896703 .905685 I.IO41365 I.34917 .741195 50 20 .673443 1.4849073 .910994 1.0977020 1.35274 .739239 40 30 .675590 1. 4801872 .916331 I.O913085 1.35634 .737277 30 40 .677732 1.4755095 .921697 1.0849554 1-35997 .735309 20 50 .679868 1.4708736 .927091 1.0786423 1.36363 .733335 10 43 ex .681998 1.4662792 .932515 1.0723687 1.36733 .731354 47 10 .684123 I.4617257 .937968 I.066134I 1-37105 .729367 50 20 .686242 I.4572127 .943451 I.0599381 1.37481 .727374 40 30 .688355 1.4527397 .948965 I.O53780I 1.37860 .725374 30 40 .690462 1.4483063 .954508 1.0476598 1.38242 .723369 20 50 .692563 1.4439 1 20 .960083 I.O415767 1.38628 .721357 10 44 .694658 1.4395565 .965689 1.0355303 1.39016 .719340 46 10 .696748 1.4352393 .971326 1.0295203 1.39409 .717316 50 20 .698832 1.4309602 .976996 I.O235461 1.39804 .715286 40 30 . 700909 1. 4267182 .982697 i.oi 76074 1.40203 .713251 30 40 .702981 I.4225134 .988432 I.OII7088 1.40606 .711209 20 i 50 1 .705047 1.4183454 .994199 1.0058348 1.41012 .709161 10 45 .707107 I.4142136 1. 000000 1. 0000000 1.41421 .707107 45 Cosine. SecanL Cotangent. Tangent Cosecant. Sine. Angle. ' no MATHEMATICAL TABLES. TABLE No. IX.— LOGARITHMIC SINES, COSINES, TANGENTS, AND COTANGENTS OF ANGLES FROM o^ TO 90^ Advancing by ic/, or one-sixth of a degree. Angle. Sine. Tangent Cotangent. . Cosine. 0° 0.000000 0.000000 Infinite. 10.000000 9o« lo' 7.463726 7.463727 ".536273 9.999998 50' 20 7.764754 7.764761 12.235239 9.999993 40 30 7.940842 7.940858 12.059142 9.999983 30 40 8.065776 8.065806 11.934194 9.999971 20 50 8.162681 8.162727 11.837273 9.9999S4 10 I 8.241855 8.24192I 11.758079 9.999934 89 10 8.308794 8.308884 11.691116 9.999910 50 20 8.366777 8.366895 11.633105 9.999882 40 30 8.417919 8.418068 11.58x932 9.999851 30 40 8.463665 8.463849 11.536151 9.999816 20 50 8.505045 8.505267 11.494733 9.999778 10 2 8.542819 8.543084 11.456916 9.999735 88 10 8.577566 8.577877 11.422123 9.999689 50 20 8.609734 8.610094 11.389906 9.999640 40 30 8.639680 8.640093 11.359907 9.999586 30 40 8.667689 8.668160 11.331840 9.999529 20 50 8.693998 8.694529 11.305471 9.999469 10 3 8.718800 8.719396 11.280604 9.999404 87 10 8.742259 8.742922 11.257078 9.999336 50 20 8. 7645 II 8.765246 11.234754 9.999265 40 30 8.785675 8.786486 11.213514 9.999189 30 40 8.805852 8.806742 11.193258 9.999110 20 SO 8.825130 8.826103 11.173897 9.999027 10 4 8.843585 8.844644 11.155356 9.998941 86 10 8.861283 8.862433 II. 137567 9.998851 50 20 8.878285 8.879529 1 1. 1 2047 1 9.998757 40 30 8.894643 8.895984 11.104016 9.998659 30 40 8.910404 8.9I1846 11.088154 9.998558 20 SO 8.925609 8.927156 11.072844 9.998453 10 5 8.940296 8.941952 11.058048 9.998344 85 10 8.954499 8.956267 11.043733 9.998232 50 20 8.968249 8.970133 11.029867 9.9981 16 40 30 8.981573 8.983577 II. 016423 9.997996 30 40 8.994497 8.996624 11.003376 9.997872 20 so 9.007044 9.009298 10.990702 9.997745 10 Cosine. Cotangent. Tangent Sine. Angle. LOGARITHMIC SINES, TANGENTS, &C. Ill Aogie. 1 1 Sine. 1 Tangent Cotangent. Cosine. 6^ 9.019235 9.021620 10.978380 9.997614 84° lo' 9,031089 9.O33JS09 10.966391 9.997480 50' 20 9.042625 9.045284 10.954716 9.997341 40 30 9-053859 9.056659 10.943341 9.997199 30 40 9.064806 9.067752 10.932248 9.997053 20 50 9.075480 9.078576 10.921424 9.996904 10 7 9.085894 9.089144 10.910856 9.996751 83 10 9.096062 9.099468 10.900532 9.996594 50 20 9.105992 9.109559 10.890441 9.996433 40 30 9. 1 15698 9.II9429 10.880571 9.996269 30 40 1 9-125187 9.129087 10.870913 9.996100 20 50 9.134470 9138542 10.861458 9.995928 10 8 9-I43SS5 9.147803 10.852197 9.995753 82 10 9-152451 9.156877 10.843123 9.995573 50 20 9.161164 9-165774 10.834226 9.995390 40 30 9.169702 9.174499 10.825501 9.995203 30 40 9.178072 9-183059 IO.81694I 9.995013 20 50 9.186280 9.I91462 10.808538 9.994818 10 9 9-194332 9199713 10.800287 9.994620 81 10 9.202234 9.207817 10.792183 9.994418 50 20 9.209992 9.215780 10.784220 9.994212 40 30 9.217609 9.223607 10.776393 9.994003 30 40 9.225092 9.231302 10.768698 9.993789 20. 50 9.232444 9.238872 IO.761128 9-993572 10 10 9.239670 9.246319 10.753681 9.993351 80 10 9.246775 9.253648 10.746352 9.993127 50 20 ' 9-253761 9.260863 10.739137 9.992898 40 3<^ 9.260633 9.267967 10.732033 9.992666 30 40 9-2673,95 9.274964 10.725036 9.992430 20 50 9.274049 9.281858 IO.718142 9.992190 10 II 9.280599 9.288652 10.71 1348 9.991947 79 10 9.287048 9.295349 10.704651 9.991699 50 20 9-293399 9-301951 10.698049 9.991448 40 30 9-299655 9.308463 10.691537 9.991 193 30 40 9-305819 9.314885 10.6851 15 9.990934 20 50 9-31 1893 9.321222 10.678778 9.990671 10 12 9.317879 9.327475 10.672525 9.990404 78 10 9.323780 9-333646 10.666354 9.990134 50 20 9-329599 9.339739 10.660261 9.989860 40 30 9-335337 9.34575s 10.654245 9.989582 30 40 9-340996 9.351697 10.648303 9.989300 20 50 9-346779 9.357566 10.642434 9.989014 10 Cosine. Cotangent. Tangent Sine. Angle. 112 MATHEMATICAL TABLES. Angle. Sine. Tangent. Cotangent. Cosine. 1 13° 9.352088 9.363364 10.636636 9.988724 If 10' 9-357524 9-369094 10.630906 9.988430 so' 20 9.362889 9.374756 10.625244 9-988133 40 30 9.368185 9380354 10.619646 9.987832 30 40 9-373414 9.385888 IO.614II2 9.987526 20 50 9-378577 9.391360 10.608640 9.987217 10 14 9-383675 9.396771 10.603229 9.986904 76 10 9.388711 9.402124 10.597876 9.986587 50 20 9-393685 9.407419 10.592581 9.986266 40 30 9.398600 9.412658 10.587342 9.985942 30 40 9-403455 9.417842 10.582158 9.985613 20 50 9.408254 9.422974 10.577026 9.985280 10 15 9.412996 9.428052 10.571948 9-984944 75 10 9.417684 9.433080 10.566920 9.984603 50 20 9.422318 9438059 IO.56194I 9-984259 40 30 9.426899 9.442988 10.557012 9.9839 1 1 30 40 9.431429 9.447870 10.552130 9-983558 20 50 9-435908 9.452706 10.547294 9.983202 10 16 9440338 9-457496 10.542504 9.982842 74 10 9.444720 9.462242 10.537758 9.982477 50 20 9.449054 9.466945 10.533055 9.982109 40 30 9-453342 9.471605 10.528395 9-981737 30 40 9-457584 9.476223 10.523777 9.981361 20 50 9.461782 9.480801 IO.519199 9.980981 10 17 9-465935 9.485339 1 0.5 1 466 1 9.980596 73 10 9.470046 9-489838 IO.51O162 9.980208 50 20 9-4741 15 9-494299 10.505701 9.979816 40 30 9.478142 9.498722 10.501278 9-979420 30 40 9.482128 9.503109 10.496891 9.979019 20 50 9.48607s 9.507460 10.492540 9.978615 10 18 9.489982 9.5II776 10.488224 9.978206 72 10 9.493851 9-516057 10.483943 9-977794 50 20 9.497682 9.520305 10.479695 9-977377 40 30 9.501476 9.524520 10.475480 9-976957 30 40 9.505234 9.528702 10.471298 9-976532 20 50 9-508956 9.532853 10.467147 9.976103 10 19 9.512642 9.536972 10.463028 9.975670 71 10 9.516294 9.541061 10.458939 9.975233 50 20 9.519911 9.545II9 10.454881 9.974792 40 30 9.523495 9.549149 10.450851 9.974347 30 40 9.527046 9.553149 10.446851 9.973897 20 50 9.530565 9.55712I 10.442879 9-973444 10 Cosine. Cotangent Tangent. Sine. Angle. LOGARITHMIC SINES, TANGENTS, &C "3 Ai«le. Sine. Tangent. Cotangent. Cosme. 20^ 9-534052 9.561066 10.438934 9.972986 700 10' 9537507 9.564983 10.435017 9.972524 50' 20 9540931 9.568873 IO.43II27 9.972058 40 30 9-544325 9.572738 10.427262 9.971588 30 40 9.547689 9.576576 10.423424 9.971113 20 50 9-551024 9.580389 IO.41961I 9.970635 10 21 9-554329 9.584177 10.415823 9.970152 69 10 9.557606 9.587941 10.412059 9.969665 50 20 9.56085s 9.591681 10.408319 9.969173 40 30 9.564075 9.595398 10.404602 9.968678 30 40 9.567269 9.599091 10.400909 9.968178 20 50 9.570435 9.602761 10.397239 9.967674 10 22 9-573575 9.606410 10.393590 9.967166 68 10 9.576689 9.610036 10.389964 9.966653 50 20 9.579777 9.61364I 10.386359 9.966136 40 30 9.582840 9.617224 10.382776 9-965615 30 40 9.585877 9.620787 10.379213 9.965090 20 50 9.588890 9.624330 10.375670 9.964560 10 23 9.591878 9.627852 10.372148 9.964026 67 10 9.594842 9.631355 10.368645 9.963488 50 20 9.597783 9.634838 10.365162 9.962945 40 30 9.600700 9.638302 10.361698 9.962398 30 40 9.603594 9.641747 10.358253 9.961846 20 50 9.606465 9.645174 10.354826 9.961290 10 M 9.609313 9.648583 IO.351417 9.960730 66 10 9.6 1 2 140 9.651974 10.348026 9.960165 SO 20 9.614944 9.655348 10.344652 9-959596 40 30 9.617727 9.658704 10.341296 9-959023 30 40 9.620488 9.662043 10.337957 9.958445 20 50 9.623229 9.665366 10.334634 9.957863 10 25 9.625948 9.668673 10.331328 9.957276 65 10 9.628647 9.671963 10.328037 9.956684 50 20 9.631326 9.675237 10.324763 9.956089 40 30 9.633984 9.678496 10.321504 9-955488 30 40 9.636623 9.681740 10.318260 9-954883 20 50 9.639242 9.684968 10.315032 9-954274 10 26 9.641842 9.688182 IO.3I1818 9-953660 64 10 9.644423 9-691381 10.308619 9-953042 SO 20 9.646984 9.694566 10.305434 9.952419 40 30 9.649527 9.697736 10.302264 9-951791 30 40 9.652052 9.700893 10.299107 9-95"S9 20 50 9.654558 9.704036 10.295964 9.950522 10 Cosine. Cotangent. Tangent Sine. Angle. 8 "4 MATHEMATICAL TABLES. Angle. Sine. Tangent. Cotangent. Cosine. 27° 9.657047 9.707166 10.292834 9.949881 63^ 10' 9-659517 9.710282 10.289718 9-949235 50' 20 9.661970 9.713386 10.286614 9.948584 40 30 9.664406 9.716477 10.283523* 9.947929 30 40 9.666824 9-719555 10.280445 9.947269 20 50 9.669225 9.722621 10.277379 9.946604 10 28 9.671609 9.725674 10.274326 9-945935 62 10 9-673977 9.728716 10.271284 9.945261 50 20 9.676328 9-731746 10.268254 9.944582 40 30 9.678663 9-734764 10.265236 9-943899 30 40 9.680982 9-737771 10.262229 9.943210 20 50 9.683284 9.740767 10.259233 9-942517 10 29 9-685571 9-743752 10.256248 9.941819 61 10 9.687843 9.746726 10.253274 9.941117 50 20 9.690098 9.749689 IO.2503II 9.940409 40 30 9.692339 9.752642 10.247358 9.939697 30 40 9.694564 9-755585 10.244415 9.938980 20 50 9.696775 9.758517 10.241483 9-938258 10 30 9.698970 9.761439 10.238561 9-937531' 60 10 9.70II51 9.764352 10.235648 9.936799 50 20 9-703317 9-767255 10.232745 .9.936062 40 30 9.705469 9.770148 10.229852 9-935320 30 40 9.707606 9.773033 10.226967 9-934574 20 SO 9.709730 9-775908 10.224092 9.933822 10 31 9.711839 9-778774 10.221226 9.933066 59 10 9-713935 9.781631 10.218369 9.932304 50 20 9.716017 9-784479 IO.21552I 9931537 40 30 9-718085 9-787319 IO.212681 9.930766 30 40 9.720140 9.790151 10.209849 9.929989 20 50 9.722181 9-792974 10.207026 9.929207 10 32 9.724210 9.795789 10.2042 1 1 9.928420 58 10 9.726225 9.798596 10.201404 9.927629 50 20 9.728227 9.801396 10.198604 9.926831 40 30 9.730217 9.804187 IO.I95813 9.926029 30 40 9732193 9.806971 10.193029 9.925222 20 50 9-734157 9.809748 10.190252 9.924409 10 33 9.736109 9.812517 10.187483 9-923591 57 10 9.738048 9.815280 10.184720 9.922768 50 20 9-739975 9.818035 IO.181965 9.921940 40 30 9.741889 9.820783 IO.I79217 9.921107 30 40 9-743792 9-823524 10.176476 9.92026^ 20 50 9.745683 9.826259 IO.I7374I 9.919424 10 Cosine. Cotangent. Tangent. Sine. Angle. LOGARITHMIC SINES, TANGENTS, &C 115 Angle. Sine. Tangent Cotangent Cosine. ' 34 9.747562 9.828987 IO.1710I3 9.918574 56° , 10' 9.749429 9.831709 . IO.16829I 9.917719 50' 20 9.751284 9-834425 10.165575 9.916859 40 30 9753128 9-837134 10.162866 9-915994 30 40 9.754960 9.839838 IO.160162 9-915123 20 50 9.756782 9-842535 10.157465 9.914246 10 35 9-758591 9.845227 10.154773 9.913365 55 10 9.760390 9.847913 10.152087 9.912477 50 20 9.762177 9-850593 10.149407 9.91 1584 40 30 9-763954 9.853268 10.146732 9.910686 30 40 9.765720 9-855938 10.144062 9.909782 20 50 9-767475 9.858602 IO.I41398 9.908873 10 36 9.769219 9.861261 10.138739 9.907958 54 10 9.770952 9-863915 10.136085 9.907037 50 20 9.77267s 9.866564 10.133436 9.9061 11 40 30 9-774388 9.869209 IO.I3079X 9.905179 30 40 9.776090 9.871849 IO.I28151 9.904241 20 50 1 9-777781 9.874484 IO.I25516 9.903298 10 1 37 9-779463 9.877114 10.122886 9.902349 53 10 1 9-781134 9.879741 10.120259 9.901394 50 20 1 9-782796 9.882363 IO.II7637 9.900433 40 30 9.784447 9.884980 10. 1 1 5020 9.899467 30 40 9.786089 9.887594 10. 1 12406 9-898494 20 50 9.787720 9.890204 10.109796 9.897516 10 38 9.789342 9.892810 IO.IO719O . 9.896532 52 10 9-790954 9.895412 10.104588 9.895542 50 20 9792557 9.898010 IO.IOI99O 9.894546 40 30 9-794150 9.900605 10.099395 9.893344 30 40 9-795733 9.903197 10.096803 9.892536 20 50 9.797307 9-905785 10.094215 9.891523 10 39 9-798872 9.908369 IO.O9163I 9.890503 51 10 9.800427 9.910951 10.089049 9.889477 50 20 9.801973 9-913529 10.086471 9.888444 40 1 3<^ 9.803511 9.916104 10.083896 9.887406 30 1 ^^ 40 9.805039 9.918677 10.081323 9.886362 20 50 9-806557 9.921247 10.078753 9.885311 10 40 9.808067 9.923814 10.076186 9.884254 50 10 9.809569 9.926378 10.073622 9.883191 50 20 9.811061 9.928940 10.071060 9.88212I 40 30 9.812544 9-931499 10.068501 9.881046 30 40 9.814019 9-934056 10.065944 9.879963 20 50 9-815485 9.93661 1 10.063389 9.878875 10 Coune. Cotangent. T.ingcnt. Sine. Angle. .:j,-«rrf'^f-" ri6 MATHEMATICAL TABLES. Angle. Sine. Tangent Cotangent. Cosine. 1 41° 9.816943 9-939163 10.060837 9.877780 49° 10' 9.818392 9-941713 10.058287 9.876678 50' 20 9.819832 9.944262 10.055738 9-875571 40 30 9.821265 9.946808 10.053192 9.874456 30 40 9.822688 9-949353 10.050647 9.873335 20 50 9.824104 9.951896 10.048104 9.872208 10 42 9-825511 9-954437 10.045563 9.871073 48 10 9.826910 9-956977 10.043023 9.869933 50 20 9.828301 9-959516 10.040484 9.868785 40 30 9.829683 9.962052 10.037948 9.867631 30 40 9.831058 9.964588 10.035412 9.866470 20 50 9-832425 9.967123 10.032877 9.865302 10 43 9-833783 9.969656 10.030344 9.864127 47 10 9-835134 9,972188 10.027812 9.862946 50 20 9.836477 9.974720 10.025280 9-861758 40 30 9.837812 9.977250 10.022750 9.860562 30 40 9.839140 9.979780 10.020220 9.859360 20 50 9.840459 9.982309 10.01 7691 9.85815I 10 44 9.841771 9-984837 IO.OI5163 9-856934 46 10 9.843076 9-987365 10.012635 9.855711 SO 20 9.844372 9.989893 IO.OIOIO7 9.854480 40 30 9.845662 9.992420 10.007580 9-853242 30 40 9.846944 9.994947 10.005053 9-851997 20 SO 9.848218 9-997473 10.002527 9850745 10 45 9.849485 10.000000 10.000000 9.849485 45 Cosine. Cotangent. Tangent. Sine. Angle. RHUMBS, OR POINTS OF THE COMPASS. 117 TABLE No. X.— RHUMBS, OR POINTS OF THE COMPASS. Points. H >i Va- I 2 3 sH 4 4H 6^ 8 Angles. 2^48' 45' 5 37 30 8 26 15 II 15 o U 3 45 16 52 30 19 41 15 22 30 o 25 18 45 28 7 30 30 56 15 33 45 o 36 33 45 39 22 30 42 n 15 45 o o 47 48 45 50 37 30 53 26 15 56 15 o 59 3 45 61 52 30 64 41 15 67 30 o 70 18 45 73 7 30 75 56 15 78 45 o 81 ZZ 45 84 22 30 87 II 15 90 o o NORTH. N ^ E N 5^ - /* E N ^ E N by E Nby E ;^ N by E J^ E N by E ^ E NNE NNE ^ E NNE ^ E NNE ^ E NE by N NE^N '' N N NE>52 NE NE J^ E NE J^ E NE J^ E NE by E ENE ^ N ENE y2 N ENE ]/{ N ENE ENE ^ E ENE ^ E ENE y^ E Eby N E% N E li N E^N EAST. NORTH. SOUTH. N ^ W N ^ W N J^ W N by W N by w ^ w N by w J^ w N by w ^ w NNW NNW j5^ W NNW j5 W NNW y^ W NW by N NW ^ N NW i^ N NW^ N NW. NW l^ W NW J^ W NW J^ W NW by w WNW y^ N WNW ^ N WNW J^ N WNW WNW ^ W WNW J^ W WNW y^ W . w by N W^ N w J4 N W 5j( N WEST. S )4 E S >^ E S^E s by E E sby E s by E ^ E s by £ )^ E SSE SSE ^ E SSE Yi E SSE J^ E SE by s SE ^ S SE ^ S SE ^ S SE SE l^ E SE ^ E SE ^ E SE by E ESE ^ S ESE y2 S ESE^ S ESE ESE % E ESE J4 E ESE y^ E E by S E ^ S E^ S E j^ S EAST. SOUTH. S s s^w s>^ w s^w s by w sby w ^ w by w J^ w by w ^ w ssw ssw ^ w ssw y^ w ssw y^ w sw by s sw^ s sw ^ s sw i^ s sw sw^ w sw J4 w sw ^ w swby w wsw y^ s wsw J4 s wsw^ s wsw wsw 5^ w wsw J^ w wsw y^ w w by s w^ s w J4s w j^ s WEST. MATHEMATICAL TABLES. TABLE No. XL— RECIPROCALS OF NUMBERS RECIPROCALS OF NUMBERS. 119 No. 57 58 59 [60 61 62 63 64 66 67 68 69 70 71 72 73 74 75 76 77 73 79 80 81 82 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 200 201 I Reciprocal. 006369 006329 006289 006250 0062 I I 006173 006135 006098 006061 006024 005988 005952 005917 005882 005848 005814 005780 005747 005714 005682 005650 005618 005587 005556 005225 005495 005464 005435 005405 005376 005348 005319 005291 005263 005236 005208 OO5181 005155 005128 005102 005076 005051 005025 005000 004975 No. 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232. 233 234 235 236 237 238 239 240 241 242 243 244 245 246 ReciprocaL 004950 004926 004902 004878 004854 00483 I 004808 004785 004762 004739 0047 I 7 004695 004673 00465 1 004630 004608 004587 004566 004545 004525 004505 004484 004464 004444 004425 004405 004386 004367 004348 004329 004310 004292 0042 74 004255 004237 004219 004202 004184 004167 004149 004132 004115 004098 004082 004065 No. 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 Reciprocal. I 004049 004032 004016 004000 003984 003968 003953 003937 003922 003906 003891 003876 003861 003846 003831 003817 003802 003788 003774 003759 003745 003731 003717 003704 003690 003676 003663 003650 003636 003623 003610 003597 003584 003571 003559 003546 003534 003522 003509 003497 003484 003472 003460 003448 003436 No. 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 332 333 334 335 336 Reciprocal. .003425 .003413 .003401 .003390 .003378 .003367 .003356 .003344 .003333 .003322 .003311 .003301 .003289 .003279 .003268 .003257 .003247 .003236 .003226 .003215 .003205 .003195 .003185 .003175 .003165 .003155 .003145 .003135 .003125 .003115 .003106 .003096 .003086 .003077 .003067 .003058 .003049 .003040 .003030 .003021 .003012 .003003 .002994 .002985 .002976 I 1 20 MATHEMATICAL TABLES. No. 337 338 339 380 381 Reciprocal. .002967 .002959 .002950 340 .002941 341 .002933 342 .002924 343 .002915 344 .002907 345 .002899 346 .002890 347 .002882 348 .002874 349 .002865 350 .002857 35J .002849 352 .002841 353 .002833 354 .002825 355 .002817 356 .002809 357 .002801 358 .002793 359 .002786 360 .002778 361 .002770 362 .002762 363 .002755 364 .002747 365 .002740 366 .002732 367 .002725 368 .002717 369 .002710 370 .002703 371 .002695 372 .002688 373 .002681 374 .002674 375 .002667 376 .002660 377 .002653 378 .002646 379 .002639 .002632 .00^625 No. 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 Redprocal. .002618 .002611 .002604 .002597 .002591 .002584 .002577 .002571 .002564 .002558 .002551 .002545 .002538 .002532 .002525 .002519 .002513 .002506 .002500 .002494 .002488 .002481 .002475 .002469 .002463 .002457 .002451 .002445 .002439 .002433 .002427 .002421 .002415 .002410 .002407 .002398 .002392 .002387 .002381 .002375 .002370 .002364 .002358 .002353 .002347 No. 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 Reciprocal. .002342 .002336 .002331 ,002326 .002320 .002315 .002309 .002304 .002299 .002294 .002288 .002283 .002278 .002273 .002268 .002262 .002257 .002252 .002247 .002242 .002237 .002232 .002227 .002222 .002217 .002212 .002208 .002203 .002198 .002193 .002188 .002183 .002179 .002174 .002169 .002165 .002160 .002155 .002151 .002146 .002141 ,002137 .002132 .002128 .002123 No. 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 5x5 516 Reciprocal. .002119 .002114 .002110 .002105 .002101 .002096 .002092 .002088 .002083 .002079 .002075 .002070 .002066 .002062 .002058 .002053 .002049 .002045 .002041 .002037 .002033 .002028 .002024 .002020 .002016 .002012 .002008 .002004 .002000 .001996 .001992 .001988 .001984 .001980 .001976 .001972 .001969 .001965 .001961 .001957 .001953 .001949 .001946 .001942 .001938 RECIPROCALS OF NUMBERS. 121 Ko. 518 520 521 522 526 528 529 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 RcaprocaL .001934 .001931 .001927 .001923 .001919 .001916 .001912 .001908 .001905 .001901 .001898 .001894 .001890 .001887 .001883 .001880 .001876 .001873 .001869 .001866 .001862 .001859 .001855 .001852 .001848 .001845 .001842 .001838 .001835 .001832 .001828 .001825 .001821 .001818 .001815 .001812 .001808 .001805 .001802 .001799 .001795 .001792 .001789 .001786 .001783 No. 562 563 564 565 566 567 568 569 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 RedprocaL .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 .00 ,00 .00 .00 .00 .00 .00 .00 .00 .00 .00 79 76 73 70 67 64 61 57 54 51 48 45 42 39 36 33 30 27 24 21 18 15 12 09 06 04 01 698 695 692 689 686 684 681 678 675 672 669 667 664 661 658 656 653 650 No. 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 Reciprocal. .001647 .001645 .001642 .001639 .001637 .001634 .001631 .001629 .001626 .001623 .001621 .001618 .001616 .001613 .001610 .001608 .001605 .001603 .001600 .001597 .001595 .001592 .001590 .001587 .001585 .001582 .001580 .001577 •001575 .001572 .001570 .001567 .001565 .001563 .001560 .001558 .001555 •001553 .001550 .001548 .001546 .001543 .001541 .001538 .001536 No. Reciprocal. 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 .001534 .001531 .001529 .001527 .001524 .001522 .001520 .001517 .001515 .001513 .001511 .001508 .001506 .001504 .001502 .001499 .001497 .001495 .001493 .001490 .001488 .001486 .001484 .001481 .001479 .001477 .001475 .001473 .001471 .001468 .001466 .001464 .001462 .001460 .001458 .001456 .001453 .001451 .001449 .001447 .001445 .001443 .001441 .001439 .001437 122 MATHEMATICAL TABLES. No. Reciprocal. 697 .001435 698 .001433 699 .001431 rOO .001429 roi .001427 '02 .001425 '03 .001422 ^04 .001420 ro5 .001418 '06 .001416 07 .001414 ro8 .001412 r09 .001410 10 .001408 11 .001406 12 .001404 13 .001403 14 .001401 15 .001399 16 .001397 17 .001395 18 .001393 19 .001391 '20 .001389 '21 .001387 22 .001385 23 .001383 24 .001381 25 .001379 26 .001377 27 .001376 28 .001374 29 .001372 ^30 .001370 '31 .001368 '32 .001366 '33 .001364 '34 .001362 '35 .001361 36 .001359 r37 .001357 38 .001355 ^39 .001353 r40 .001351 I '41 .001350 ! No. Reciprocal. 742 .001 743 .001 744 .001 745 .001 746 .001 747 .001 748 .001 749 .001 750 .001 751 .001 752 .001 753 .001 754 .001 755 .001 756 .001 757 .001 758 .001 759 .001 760 .001 761 .001 762 .001 763 .001 764 .001 765 .001 766 .001 767 .001 768 .001 769 .001 770 .001 771 .001 772 .001 773 .001 774 .001 775 .001 776 .001 777 .001 778 .001 779 .001 780 .001 781 .001 782 .001 783 .001 784 .001 785 .001 786 .001 348 346 344 342 340 339 337 335 333 332 328 326 325 323 321 319 318 316 314 312 311 309 307 305 304 302 300 299 297 295 294 292 290 289 287 285 284 282 280 279 277 276 274 272 No. 787 788 789 830 831 Reciprocal. .001271 .001269 .001267 790 .001266 791 .001264 792 .001263 793 .001261 794 .001259 795 .001258 796 .001256 797 .001255 798 .001253 799 .001251 800 .001250 801 .001248 802 .001247 803 .001245 804 .001244 805 .001242 806 .001241 807 .001239 808 .001238 809 .001236 Bid .001235 811 .001233 812 .001232 813 .001230 814 .001229 815 .001227 816 .001225 817 .001224 818 .001222 819 .OOI22I 820 .001220 821 .001218 822 .001217 823 .001215 824 .001214 825 .001212 826 .001211 827 .001209 828 .001208. 829 .001206 .001205 .001203 No. 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 Reciprocal. .00: .00] .00: .00: .00: .00: .00: .00: .00: .00: .00: .00: .00; .00: .00; .00; .00; .00; .00] .00: .00] .00: .OOj .00: .00; .00: .00: .00: .00; .00: .OOJ .OOJ .00; .00: .00; .00: .00 .00: .00: .00; .00: .00: .00 .00; .00: 202 200 199 198 196 195 193 192 190 189 188 186 185 183 182 181 179 178 176 175 174 172 171 I 70 168 167 166 164 ^^3 161 160 159 157 156 155 153 152 151 149 148 147 145 144 143 142 RECIPROCALS OF NUMBERS. 123 Now Rec ip io ca L 877 .001 878 .001 879 .OOI 880 .001 S8I .001 SS2 .001 883 .001 884 .001 885 .001 886 .001 887 .OOI 888 .001 889 .001 Sgo .001 891 .001 892 .001 893 .001 894 .001 895 .001 896 .001 897 .001 898 .001 899 .001 QOO .001 901 .001 902 .001 903 .001 904 .001 905 .001 906 .001 907 .001 140 139 138 136 135 134 133 131 130 129 127 126 125 124 122 121 120 119 118 116 "5 114 112 III IIO 109 107 106 104 103 No. 908 909 910 911 912 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 931 932 933 934 935 936 937 938 Reciprocal. .001101 .001100 .001099 .001098 .001096 .001095 .001094 .001093 .001092 .001091 .001089 .001088 .001087 .001086 .001085 .001083 .001082 .001081 .001080 .001079 .001078 .001076 .001075 .001074 .001073 .001072 .001071 .001070 .001068 .001067 .001066 No. 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958. 959 960 961 962 963 964 965 966 967 968 969 RcoprocaL .001065 .001064 .001063 .001062 .001060 .001059 .001058 .001057 .001056 .001055 .001054 .001053 .001052 .001050 .001049 .001048 .001047 .001046 .001645 .001044 .001043 .001042 .001041 .001040 ,001038 .001037 .001036 .001035 .001034 .001033 .001032 No. 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 Reciprocal. ,001031 .001030 .001029 .001028 .001027 .001026 .001025 .001024 .001022 .001021 .001020 .001019 .001018 .001017 .001016 .001015 .001014 .001013 .001012 .001011 .001010 .001009 .001008 .001007 .001006 .001005 .001004 .001003 .001002 .001001 .001000 WEIGHTS AND MEASURES. WATER AND AIR AS STANDARDS FOR WEIGHT AND MEASURE. WATER AS A STANDARD. There are four notable temperatures for water, namely, 32° R, or 0° C. = the freezing point, under one atmosphere. 3 9°. I or 4° = the point of maximum density. 62° or 1 6°. 66 = the British standard temperature. 212° or 100° = the boiling point, under one atmosphere. The temperature 62° F. is the temperature of water used in calculating the specific gravity of bodies, with respect to the gravity or density of water as a basis, or as unity. In France, the temperature of maximum density, 39°.! F., or 4° C, is used for this purpose, for solids. Weight of one cubic foot of Pure Water, At 32° F. =■ 62.418 pounds. At 39°. I = 62.425 „ At 62° (Standard temperature) = 62.355 » At 212° = 59.640 „ The weight of a cubic foot of water is, it may be added, about 1000 ounces (exactly 998.8 ounces), at the temperature of maximum density. The weight of water is usually taken in round numbers, for ordinary calculations, at 62.4 lbs. per cubic foot, which is the weight at 52^.3 F. ; or it is taken at 62)^ lbs. per cubic foot, where precision is not required, equal to 1^^ lbs. The weight of a cylindrical foot of water at 62° F. is 48.973 pounds. Weight of one cubic inch of Pure Water, At 32® F. = .03612 pounds, or 0.5779 ounce. At39°.i =.036125 „ ,,0.5780 „ At 62° =.03608 „ ,,0.5773 „ or 252.595 grains. At 212** = .03451 „ „ 0.5522 „ The weight of one cylindrical inch of pure water at 62** F. is -02833 pound, or 0.4533 ounce. WATER AND AIR AS STANDARDS. 1 25 Volume of one pound of Pure Water, At 32** F. = .016021 cubic foot, or 27.684 cubic inches. At 39^1 = .016019 „ „ 27.680 „ At 62"* = .016037 „ „ 27.712 „ At 212** = .016770 „ „ 28.978 „ The volume of one ounce of pure water at 62** F. is 1.732 cubic inches. The Gallon. The weight of one gallon of water at the standard temperature, 62*^ F., is 10 pounds, and the correct volume is 0.160372 cubic foot, or 277.123 cubic inches. But in an Act of Parliament, now partly repealed, which came into force in 1825, the volume of one gallon is stated to be 277.274 cubic inches; this is the commonly accepted volume. The volume of 10 pounds of water at 62° F. is, therefore, to the volume of the imperial gallon, as i to 1.000545. And, the weight of an imperial gallon of water at 62° F. is 10.00545 pounds avoirdupois j or 10 pounds 38.15 grains. One cubic foot of water contams 6.2355 gallons, or approximately 6^ gaDons. The volume of water at 62*" F., in cubic inches, multiplied by .00036, giv« the capacity in gallons. The capacity of one gallon is equal to one square foot, two inches deep nearly (exactly 1.924 inches); or to one circular foot, 2j^ inches deep nearly (exactly 2.45 inches). One ton of water at 62° F. contains 224 gallons. Other Measures of Water. Volume of given weights of water, at 62.4 pounds per cubic foot: — I ton 35-90 cubic feet. I cwt 1.795 » I quarter 449 „ , r .016 cubic foot, or ' P^^^ • J 27.692 cubic inches. I ounce I-73I » I tonne, at 39°.! F 35-3I56 cubic feet. 1 -1 ..or? f -0353 cubic foot, or I kdogiamme, at 39°.i F | 61.0^5 cubic inches. I tonne, at 52;.3 F. ) 35.330 cubic feet (62.4 pounds per cubic foot) J ^^ ^^ Thirty-six cubic feet, or i^ cubic yards, of water, at 62.4 pounds per tabic foot, being at the temperature 52^.3 F., weigh about one ton (exactly 6.4 pounds more). Ctae cubic yard, or twenty-seven cubic feet, of water weighs about 15 cwt, or ^ ton (exactly 4.8 pounds more). One cubic metre of water is equal in volume to 35.3156 cubic feet, or 1.308 cubic yard, or 220.09 gallons; and, at 62.4 pounds per cubic foot, it weighs i ton nearly (exactly 36.3 pounds less). It is nearly equivalent 126 WEIGHTS AND MEASURES. to the old English tun of 4 hogsheads — 210 imperial gallons, and is a better unit for measuring sewage or water-supply than the gallon. The cubic metre is generally used on the Continent for such measurements. A pipe one yard long holds about as many pounds of water as the square of its diameter in inches (exactly 2 per cent. more). Pressure of Water. A pressure of one lb. per square inch is exerted by a column of water 2.3093 feet, or 27.71 inches high, at 62® F,; and a pressure of one atmos- phere, or 14.7 lbs. per square inch, is exerted by a column of water 33.947 feet high, or 10.347 metres, at 62° F. A column of water at 62"^ F., one foot high, presses on the base with a force of 0.433 lb., or 6.928 ounces per square inch. A column 100 feet high presses with a force of 43^ lbs. per square inch. A column one metre high presses with a force of 1.422 lbs. per square inch. A column of water one inch high, presses on the base with a force of 0.5773 ounce per square inch, or 5.196 lbs. per square foot. A column of water one mile deep, weighing 62.4 pounds per cubic foot, presses on the base with a force of about one ton per square inch (fresh water exacdy 48 lbs. more; sea- water exactly 107.5 ^^s. more). Water is hardly compressible under pressure. Experiment appears to show that for each atmosphere of pressure it is condensed 47)^ millionths of its bulk. Sea-water. One cubic foot of average sea-water, at 62** F., weighs 64 pounds, and the weight of fresh water is to that of sea-water as 39 to 40, or as i to 1.026. Thirty-five cubic feet of sea-water weighs one ton. One cubic yard of sea-water weighs i^yi cwt nearly (8 lbs. less). One cubic metre of sea-water weighs fully one ton (20 lbs. more). Average sea-water is composed as follows : — Per xoo paits. Per zoo parts. Chloride of sodium • (common salt), 2.50 Sulphuret of magnesium, 0.53 Chloride of magnesium, 0.33 Carbonate of lime, ) Carbonate of magnesia, J Sulphate of lime, o.oi Solid matter, say, 3.40 Water, 96.60 100.00 » I showing that sea-water contains ^^^th part of its weight of solid matter in solution. According to R^clus, the mean specific gravity of sea-water is 1.028. In the Mediterranean Sea, it is 1.029; in the Black Sea, 1.016. The mean quantity of salts, or solid matter, in solution, is 3.44 percent., three-fourths of which is common salt In the Red Sea, the water contains 4.3 per cent. • in the Baltic Sea, 5 per cent. ; and at Cronstadt, 2 per cent WATER AND AIR AS STANDARDS. 12/ Ice and Snow, One cubic foot of ice at 32° F. weighs 57.50 lbs. One pound of ice at 32° F. has a volume of .0174 cubic foot, or 30.067 cubic inches. The volume of water at 32° F. is to that of ice at 32** F., as i.ooo to 1.0855; ^^ expansion in passing into the sohd state being above 8)^ per cent of the volume of water. The specific density of ice is 0.922, that of water at 62° F. being = i. The melting point of ice is ^2"^ F., or 0° C, under the ordinary atmos- pheric pressure, of 14.7 lbs. per square inch. Under greater pressure the melting point is lower, being 'at the rate of .0133° F. for each additional atmosphere of pressure. The specific heat of ice is .504, that of water being = i. One cubic foot of fresh snow weighs 5.20 lbs. Snow has 12 timeis the bulk of water, and its specific gravity is .0833. French and English Measures of Water, One litre of water is equal to 0.2201 gallon, or 1.761 pints: about i|:^ pints. One gallon is equal to 4.544 litres, and one pint is .568 litre. One litre of water at 39'^.! F., or 4° C, the temperature of maximum density, weighs one kilogramme, or 2.2046 lbs.; at the temperature 62° F., or 1 6°. 7 C, it weighs 2.202 lbs. looo litres = one cubic metre, equal to 35.3156 cubic feet; and, at 39^1 F., or 4** C, weigh 1000 kilogrammes, or one ton nearly (35.4 lbs. less). AIR AS A STANDARD. The mean pressure of the atmosphere at the level of the sea, is equal to 14.7 lbs. per square inch, or* 21 16.4 lbs. per square foot; or to 1.0335 kilogrammes per square centimetre. This is called one atmosphere of pressure. The following are measures of pressures (see also pages 1 45, 158): — One atmosphere of pressure : — (i.) A column of air at 32° F., 27,801 feet, or about 5j^ miles high, of uniform density, equal to that of air at the level of the sea. (2.) A column of mercury at 32° F., 29.922 inches or 76 centi- metres high; nearly 30 inches. At 62° F., the height is 30 inches. (3.) A column of water at 62** F., 33.947 feet or 10.347 metres high; nearly 34 feet. A pressure of i lb. per square inch: — (i.) A column of air at 32° F., 189 1 feet high, of uniform density as above. (2.) A column of mercury at 32' F., 2.035 inches or 51.7 millimetres high. At 62° F., the height is 2.04 inches. (3.) A column of water at 62° F., 2.31 feet or 27.72 inches high. A pressure of i lb. per square foot: — (i.) A column of air at 32® F., 13.13 feet high, of uniform density as above. (2.) A column of mercury at 32° F., .0141 inch or .359 millimetre high. At 62° F., the height is .01417 inch. (3.) A column of water at 62° F., .1925 inch high. The density, or weight of one cubic foot of pure air, under a pressure of one atmosphere, or 14.7 lbs. per square inch, is At 32° F., = .080728 pound, or 1.29 ounce, or 565.1 grains. At62**F., = .076097 „ „ 1.217 „ „ 53;2.7 „ The weight of a litre of pure air, under one atmosphere, at 32° F., is 1.293 gnunmesy or 19.955 grains. 128 WEIGHTS AND MEASURES. The weight of air, compared with that of water at three notable tempera- tures, and at 5 2°. 3, under one atmosphere, is as follows: — 773.2 times the weight of air at 32° F. 773-27 » » »f 772*4 » » » 819.4 „ „ 62^ 820 „ „ „ The volume of one pound of air at 32** F., and under one atmosphere of pressure, is 12.387 cubic feet. The volume at 62° F., is 13. 141 cubic feet. The specific heat of air at constant pressure is .2377, and at constant volume .1688, that of water being = i. Weight of water at : 32° F., 39". I, 62°, 62°, »> » Sa^-S. GREAT BRITAIN AND IRELAND.— IMPERIAL WEIGHTS AND MEASURES. The origin of English measures is the grain of com. Thirty-two grains of wheat, dried and gathered from the middle of the ear, weighed what was called one penn)rweight; 20 pennyweights were called one ounce, and 20 ounces one pound. Subsequently, the pennyweight was divided into 24 grains. Troy weight was afterwards introduced by William the Conqueror, from Troyes, in France; but it gave dissatisfaction, as the troy pound did not weigh so much as the pound then in use; consequently, a mean weight was established, making 16 ounces equal to one pound, and called avoir- dupois {avoir du poids), , Three grains of barleycorn, well-dried, placed end to end, made an inch — the basis of length. The length of the arm of King Henry I. was made the length of the ulna^ or ell, which answers to the modern yard. The imperial standard yard is a solid square bar of gun-metal, kept in the office of the Exchequer at Westminster, 38 inches in length, i inch square, at the temperature 62° F., composed of copper 16 ounces, tin 2^ ounces, and zinc i ounce. Two cylindrical holes are drilled half through the bar, one near each end, and the centres of these holes are 36 inches, or 3 feet, apart — the length of the imperial standard yard. Compared with a pendu- lum vibrating seconds of mean time, at the level of the sea, in the latitude of London, in a vacuum, the yard is as 36 inches in length to 39.1393 inches, the length of the pendulum. Measures of capacity were based on troy weight; it was enacted that 8 pounds troy of wheat, from the middle of the ear, well dried, should make i gallon of wine measure, and that 8 such gallons should make I bushel. The imperial gallon is now the only standard measure of capacity, and it contains 277.274 cubic inches. It is said to be the volume of 10 pounds avoirdupois of distilled water, weighed in air, at 62** F. Note, — The exact volume of 10 pounds of distilled water at 62** F. is 277.123 cubic inches. GREAT BRITAIN AND IRELAND.— LENGTH. 1 29 Tables of weights and measures are conveniently classified thus — I. Length; 2. Surface; 3. Volume; 4. Capacity; 5. Weight. The following are some of the principal units of measurement : — The acr^j for land measure. The mi/^, for itinerary measure. The yardj for measure of drapery, &c. The coomb^ for capacity of com, &c. The gallon^ for capacity of liquids. The graifij for chemical analysis. TYsQ found, for grocers* ware, &c The stone of 8 pounds, for butchers' meat The stone of 14 pounds, for flour, oatmeal, &c. I. Measures of Length. — Tables No. 12. Lineal Measure. 3 barleycorns, or'V 12 lines, or f • i. 72 pomts, or I 1000 mils / 3 inches i palm. 4 inches i hand. 9 inches v i span. 12 inches '. i foot 18 inches i cubit 3 feet I yard. 2^ feet i.military pace. 5 feet I geometrical pace. 2 yards i fathom. 5j^ yards * i rod, pole, or perch. zt^r] ^f-i-«- 8 furlongs, or "j 1760 yards, or > i mile. 5280 feet j 3 miles I league. 2240 yards, or ) j . , ., The inch is also divided into halves, quarters, eighths, and sixteenths; sometimes mto tenths. The hand is used as a measure of the height of horses. The miUtary face is the length of the ordinary step of a man. The geometrical pace is the length of two steps. A thousand of such paces were reckoned to a mile. The fathom is used in soundings to ascertain depths, and for measuring cordage and chains. 9 130 WEIGHTS AND MEASURES. Land Measure, 7.92 inches i link. 100 links, or \ ^^^^^\^^ \ X chain. 22 yards, or f 4 poles ) 10 chains i furlong. 80 chains, or ) _., 8 furlong / ^ °^^ The^, or woodland pole ox ferchy is 18 feet Tht forest pole \s 21 feet. . Nautical Measure. 6086.44 feet, or \ 1000 fathoms, or ( f i nautical mile, 10 cables, or C ( or knot. 1. 1528 statute miles ) 3 nautical miles i league. 60 nautical miles, or \ 69.168 statute miles or > i degree. 20 leagues j ( Circumference 360 degrees...' < of the earth at ( the equator. The above value of the nautical mile is that which is commonly taken, and is the length of a minute of longitude at the equator. The mean length of a minute of latitude at the mean level of the sea is nearly 6076 feet, or 1.1508 statute miles. The nautical fathom is the thousandth part of a nautical mile, and is, on an average, about ^th longer than the common fathom. Cloth Measure, 2}^ inches , i nail. 2 nails I finger-length. 4 nails, or 9 inches i quarter. 4 quarters i yard. 5 quarters i elL Wire-Gauges. The " Birmingham Wire-Gauge " is a scale of notches in the edge of a plate, of successively increasing or decreasing widths, to designate a set of arbitrary sizes or diameters of wire, ranging from about half an inch down to the smallest size easily drawn, say, four-thousands of an inch. The practical utility of such a gauge is obvious, when it is considered how far beyond the means supplied by the graduations of an ordinary scale of feet and inches is the measurement of the gradations of the wire-gauge. But the "Birming- ham Wire-Gauge" is a variable measure. The principle, if there was any, on which it was originally constructed, is not known. Mr. Latimer Clark states that, when plotted, the width? of the gauge range in a curve approxi- GREAT BRITAIN AND IRELAND. — WIRE-GAUQES. 131 mating to a logariAmic curve, such as would be found by the successive addition of 10 or 12 per cent to the width of the notches of the gauge. However that may be, there are many varieties of the wire-gauge in existence. The oldest and best-known gauge is that of which the numbers were care- fully measured by Mr. Holtzapffel, and published by him in 1847. It has been, and still is, widely followed in the manufacture of wire; and also of tubes in respect of their thickness. It gives 40 measurements ranging from .454 inch to .004 inch, and is contained in Table No. 13. Although there are only 40 marks in the table, there are 60 different sizes of wire made, for which intermediate sizes have been added to the gauge. This table haJs also been used in rolling sheet iron, sheet steel, and other materials, and for joiners' screws; but it appears to be falling into disuse for these purposes. Birmingham Wire-Gauge (HoltzapffeVs), — Table No. 13. For Wire and Tubes chiefly; and for Sheet Iron and Steel formerly. ! Maik. SlTT, Mark. Si2e. Mark. Size. Mark. Size. No. Inch. No. , Inch. No. Inch. No. Inch. 0000 .454 7 .180 17 .058 27 .016 000 .425 8 .165 18 .049 28 .014 00 .380 9 .148 19 .042 29 .013 •340 10 .134 20 •035 30 .012 I .300 II .120 21 .032 31 .010 2 .284 12 .109 22 .028 32 .009 3 -259 13 •09s 23 .025 33 .008 4 .238 14 .083 24 .022 34 .007 5 .220 15 .072 25 .020 35 .005 6 .203 16 .065 26 .018 36 .004 BiRMiNGH.^M Metai^Gauge, or Plate-Gauge (HoltzapffePs), — Table No. 14. For Sheet Metals, Brass, Gold, Silver, &c. 1 M«k. Six. Mark. Sire. Mark. Size. Mark. Size. Ko. Inch. No. Inch. No. .Inch. No. Inch. I .004 10 .024 19 .064 28 .120 2 .005 II .029 20 .067 29 .124 3 .008 12 .034 21 .072 30 .126 4 .010 13 .036 22 .074 31 .133 5 .012 14 .041 23 .077 32 .143 6 .013 15 .047 24 .082 33 .145 7 'OI5 16 •051 25 .095 34 .148 8 .016 17 .057 26 .103 35 .158 9 .019 18 .061 27 .113 36 .167 Another of HoltzapffeVs tables, No. 14, the Plate-Gauge^ has been, and may now, to some extent, be, employed for most of the sheet metals, except- 132 WEIGHTS AND MEASURES. Lancashire Gauge {Holtzapffd' s\ — Table'^o. 15. For Round Steel Wire, and for Pinion Wire. Maik. No. Size. Mark. No. Size. Mark. Sirr. Mark. No. Size. Maik. Size. Inch. Inch. No. Inch. Inch. No. Inch. 80 .013 57 .042 34 .109 II .189 M •295 79 .014 56 .044 33 .III 10 .190 N .302 78 .015 55 .050 32 •115 9 .191 .316 77 .016 54 •055 31 .118 8 .192 P .323 76 .018 53 .058 30 •125 7 .195 Q •332 75 .019 52 .060 29 .134 6 .198 R •339 74 .022 51 .064 28 .138 5 .201 S •348 73 .023 50 .067 27 .141 4 .204 T .358 72 .024 49 .070 26 ■143 3 .209 U .368 71 .026 48 •073 25 .146 2 .219 V 377 70 .027 47 .076 24 .148 I .227 w .386 69 .029 46 .078 23 .150 A .234 X ■397 68 .030 45 .080 22 .152 B .238 Y .404 67 .031 44 .084 21 .157 C .242 Z -413 66 .032 43 .086 20 .160 D .246 Ai .420 65 *^ZZ 42 .091 19 .164 E .250 Bi ■431 64 .034 41 .095 18 .167 F •257' Ci •443 63 •035 40 .096 17 .169 G .261 Di .452 62 .036 39 .098 16 .174 H .266 Ei .462 61 .038 38 .100 15 .175 I .272 Fi .475 60 .039 37 .102 14 .177 J .277 Gi ■484 59 .040 36 •105 13 .180 K .281 Hi .494 58 .041 35 .107 12 .185 L .290 ing iron and steel : as copper, brass, gilding-metal, gold, silver, and platinum. The intervals are closer or smaller than those of the wire-gauge, and the maximum size, for No. 36, is '/6 inch. When thicker sheets are wanted, their measures are sought in the Birmingham wire-gauge. The last table, No. 15, by Holtzapffel, the Lancashire Gauge, is employed exclusively for the bright steel wire prepared in Lancashire, and the steel pinion-wire for watch and clock makers. The larger sizes are marked by- capital letters, to distinguish them from the others. This, the second part of the table, is known as the Letter-Gauge, Needle- Gauge, for needle wire. The sizes correspond with some of those of the Holtzapffel wire-gauge. The following are the relative marks for equal sizes on the two gauges : — Needle wire -gauge — Nos. i, 2, 2j^, 3, 4, 5, thence to 21, corresponding to B. W.-G. — i8j^, 19, 19^, 20, 21, 22, thence to 38. * Music IVire-gauge, for the strings of pianofortes. The marks used are Nos. 6 to 20. The following are the relative marks for equal sizes with the Holtzapffel wire-gauge: — Music wire-gauge — Nos. 6, 7, 8, 9, 10,11, 12,14,16,18,20, corresponding to B. W.-G. — 26, 25)^, 25, 24}^, 24, 23 J^, 23, 22, 21, 20, 19. No. 6, the thinnest wire now used, measures about one fifty-fifth of an inch in diameter, and No. 20 about one twenty-fifth of an inch. GREAT BRITAIN AND IRELAND. — WIRE-GAUGES. 133 The preceding Tables of Gauges have been extracted from HoltzapfFers estimable work on Turning and Mechanical Majnipulation^ 1847. Messrs. Rylands Brothers, of Warrington, manufacture iron wire accord- iDg to the gauge in Table No. 16. Warrington Wire-Gauge (Rylands Brothers), — Table No. 16. Mark. Size. Mark. Size. Mark. Size. Mark. No. Size. No. Inch. No. Inch. No. Inch. Inch. 7/0 1/2 .326 8 •159 15 .069 6/0 15/32 I .300 9 .146 16 .0625, or Vx6 5/0 7/16 2 .274 TO •133 17 .053 4,^ 13/32 3 -25, or K io>^ .125, or^ 18 .047 3/0 3/8 4 .229 II .117 19 .041 2/0 11/32 5 .209 12 .io,or V,o 20 .036 6 .191 13 .090 21 •o3i5»orV33 7 •174 14 •079 22 .028 For sheets, the wire-gauge that seems to be adhered to by the iron-sheet rollers of South Staffordshire, is a scale comprising 32 measurements, ranging from .3125 inch to .0125 inch, contained in Table No. 17. Birmingham Wire-Gauge. — ^Table No. 17. For Iron Sheets chiefly. No. Size. No. Inch. 3125 (Vx6) 9 28125 10 25 (X) II 234375 12 21875 13 203125 14 1875 (V,6) 15 171875 16 Size. No. Inch. •15625 .140625 .125 (>^) 17 18 19 .1125 20 •10 (Vio) .0875 21 22 .075 .0625 (Vxe) 23 24 Size. No. Inch. .05625 25 .05 (V-o) 26 •04375 27 .0375 28 •034375 29 .03125(732) 30 .028125 31 •025 (V40) 32 Inch. .02344 .021875 .020312 .01875 .01719 .015625 .01406 .0125 (Veo) Sir Joseph Whitworth, in 1857, introduced his Standard Wire-Gauge, ranging fix)m a half inch to a thousandth of an inch, and comprising 62 measurements, as given in Table No. 18. It commences with the smallest size, and increases by thousandths of an inch up to half an inch. The smallest size, V'loooth of an inch, is No. i ; No. 2 is Vioooths of an inch, and so on, increasing up to No. 20 by intervals of Vioooth of an inch; from No. 20 to No. 40 by '/loooths; from No. 40 to No. 100 by s/ioooths of an inch. The sizes are designated or marked by their respective numbers in thousandths of an inch. It appears that the Whitworth Gauge is entering into general use ; and, in the manufacture of wire, at least, this and Rylands* gauge are likely soon to supersede the Holtzapifel scale. 134 WEIGHTS AND MEASURES. Sir Joseph Whitworth & Co.'s Standard Wire-Gauge. — ^Table No. i8. Mark. Size. Mark. Size. Mark. Size. Mark. Size. No. Inch. No. Inch. No. Inch. No. Inch. I .001 17 .017 55 .055 200 .200 2 .002 18 .018 60 .060 220 .220 3 .003 19 .019 6$ .065 240 .240 4 .004 20 .020 70 .070 260 .260 S .005 22 .022 75 .075 280 .280 6 .006 24 .024 • 80 .080 300 .300 7 .007 26 .026 85 .085 325 .325 8 .008 28 .028 90 .090 350 .350 9 .009 30 .030 95 .095 375 •375 ID .010 32 .032 100 .100 400 .400 II .Oil 34 .034 no .110 425 .425 12 .012 36 .036 120 .120 45«> .450 13 .013 3S .038 135 •135 475 .475 14 .014 40 .040 150 .150 500 .500 15 .015 45 .045 165 .165 16 .016 50 .050 180 .180 Common Fractions of an Inch and Holtzapffel's Wire-Gauge. — Table No. 19. Fraction. Inch. 'A 'A ■A 'A 'A •A ■A 'Ac 'A. •A, '/.6 •/ :^ Wire-Gauge. so 21 a« 24 25 Va6 No. fuU bare bare bare full full full bare o 3 6 8 9 I 2 3 3 4 4 bare 5 rather bare 6 6 bare 7fuU 7 bare 8 full 8 rather full 8 rather bare 8 bare 9 rather full 9 9 bare 20 full Fraction. Inch. Va8 V3« V37 V38 V39 'Ac v:: 'r Vs6 V58 V60 Wire-Gauge. No. 20 rather full 20 20 rather bare 21 rather full 21 21 rather bare 21 bare 22 full 22 rather full 22 22 bare 23 full 23 rather full 23 23 bare 24 full 24 25fuU 25 25 bare 26 rather full 26 26 bare 27 full Fraction. Wire-Gauge. Inch. V 70 V74 V76 'As Vso •As :^ /95 V :;■ Vx3o V140 Vis© VI60 180 100 zo xao V 7 aoo 950 No. 27 27 rather 27 bare 28 full 28 rather 28 28 bare 29 rather 29 29 bare 30 rather 30 bare 31 rather 31 32 33 full 33 l>are 34 34 rather 34 bare 35 Ml 35 36 bare full full full full bare GREAT BRITAIN AND IRELAND. — FRACTIONS OF INCH. 1 35 Inches and their Equivalent Decimal Values in Parts of a Foot. — Table No. 20. Inches. Fiactionoffoot Foot. I Via 08^-1 2 7 :; .1667 25 4 c 1 •3333 4167 D 6 7 ■A 7/„ ...... .Af * vr f ■5 eS-iX / 8 /la Va 3/. oo .6667 7c V 10 II /o .8333 QI67 12 • / xa I y *^/ I.O Fractional Parts of an Inch, and their Decimal Equivalents. Tables No. 21. Eighths. Eighths. Fractions. Inch. I i/g 12^ 2 X /o •A S/g ^•J 77c 4 c /<» V. S/g o# J •5 621: 6 7 /«» 7/. •*'• J •75 87s / •••••• 8 I / J 1.0 Twelfths, Twelfths. Fractions. Inch. I...... «/„ oZx%% 2 'A i/« •125 16667 3 4. 'A V, •^5 '?'?^^'? ^ 5 6. 'k •• 00000 .41667 5 7 8 /* t •5^333 66666 9 10 5/g •75 S'?^^^ II 12 /o "A. I ^0000 .91667 1.0 136 WEIGHTS AND MEASURES. Sixteenths and Thirty-seconds. — Tables No. 21 (continued). Thirty- Seconds. Sutteenths. Fractions. Inch. I 2 3 4 5 6 7 8 9 10 II 12 13 14 IS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 I 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 v„ .: .03125 .0625 •0937s "5 •15625 .1875 .21875 •*S .28125 .3125 •34375 •375 .40625 •4375 .46875 •S •53125 •5625 •59375 .625 .65625 •6875 •71875 •75 .78125 .8125 .84375 .875 .90625 •9375 .96875 1.0 /3* 3/,. / 3' 5/„ '3a • 7/„ /3« V, :'•.... /3* 3/3 »3/„ /3a »/. !{:!... »7/„ / 3* 'A6 «9/^ /3« • 91/ / 3* "A6 23/„ / 3« »s/.. '3« • a7/„ / 3* ap/,^ '3* • 'V.6 3x/„ / 3* I II. Measures of Surface. — Tables No. 22. Superficial Measure, 144 square inches, or 7 - 183.35 circular inches 3 i square foot 9 square feet i square yard loo square feet i square. 27 2 J^ square feet, or ) nH 30^ square yards j ' ^^* The square is used in measuring flooring and roofing. The rod is used in measuring brick-work. GREAT BRITAIN AND IRELAND. — SURFACE, VOLUME. 1 37 Builder^ Measurement, I superficial part i square inch. 12 parts "i inch" (12 square inches). 12 "inches" i square foot. This table is employed in the superficial or flat measure of boards, glass, stone, artificers' work, &c. Land Measure. 9 square feet i square yard. 30X square yards -j ^ ^^l^^ 16 square poles i square chain. 40 square poles, or ) ^ 1 2 10 square yards J 4 roods, or 10 square chains, or 160 square poles, or \ i acre.* 4,840 square yards, or 43,560 square feet 640 acres, or ) -1 3,097,600 squa^ yards } i square imle. 30 acres i yard of land. 100 acres i hide of land. 40 hides I barony. * The side of a square having an area of one acre is eqnal to 69.57 lineal yards. III. Measures of Volume. — Tables No. 24. Solid or Cubic Measure, 1728 cubic inches \ 2200.15 cylindrical inches ( i cubic foot 3300.23 spherical inches j 6600.45 conical inches ) 27 cubic feet i cubic yard, or load. 35.3156 cubic feet or \ ^ ^^^j^ 1.308 cubic yards J ^ate. — ^The numbers of cylindrical, spherical, and conical inches in a cubic foot, are as I, 1. 5, 3- Builderi Measurement, I solid part 12 cubic inches. 12 solid parts i "inch" (144 cubic inches). 12 "inches" i cubic foot This table is used in measuring square-sided timber, stone, &c. 138 WEIGHTS AND MEASURES. Note, — ^The cubic contents of a piece, 6 inches square and 4 feet long is i cubic foot 7 8J^ 12 17 24 3 2 I I I » I I I 2 4 Decimal Parts of a Square FogTj , IN Square Inches. — Table No. 23. Hundredth Square Hundredth Square Hundredth Square Hundredth Square Parts. Inches. Parts. Ihches. Parts. Inches. Parts. Inches. I 1.44 26 37.4 51 73.4 76 109.4 2 2.88 27 38.9 52 74.9 77 IIO.9 3 4.32 28 40.3 53 • 76.3 78 II2.3 4 5-76 29 41.8 54 77.8 79 1 13.8 5 7.20 30 43.2 55 79.2 80 II5.2 6 8.64 31 44.6 56 80.6 81 I16.6 7 10. 1 32 46.1 57 82.1 82 1 18. 1 8 "5 33 47.5 58 83.5 83 "95 9 13.0 34 49.0 59 85.0 84 121.0 10 14.4 35 50-4 60 86:4 85 122.4 II iS-8 36 51.8 61 87.8 86 123.8 12 17.3 37 53-3 62 89.3 87 125.3 13 18.7 38 54.7 63 90.7 88 126.7 14 20.2 39 56.2 64 92.2 89 128.2 . 15 21.6 40 57.6 65 93-6 90 129.6 16 23.0 41 58.0 66 95.0 91 131.0 17 24.5 42 60.5 67 96.5 92 132.5 18 259 43 61.9 68 97.9 93 133.9 19 27.4 44 63.4 69 99-4 94 135-4 20 28.8 45 64.8 70 100.8 95 136.8 21 30.2 46 66.2 71 102.2 96 138.2 22 317 47 67.7 72 103.7 97 139-7 23 33.1 48 69.1 73 105.1 98. 141. 1 24 34.6 49 70.6 74 106.6 99 142.6 25 36.0 50 72.0 75 108.0 100 144.0 IV. Measures of Capacity. — Tables No. 25. Liquid Measure, 8.665 cubic inches i gill or quartern. 4 gills (34.659 cubic inches) i pint 2 pints I quart 2 quarts i pottle. 4 quarts, or 8 pints (277.274 cubic inches) i gallon. 6.2355 gallons I cubic foot The ham-gallon^ for milk, is equal to 2 imperial gallons. GREAT BRITAIN AND IRELAND. — CAPACITY. 1 39 Dry Measure, 2 pints I quart 4quarts i gallon. 2 gallons I peck. 8S;s°'} (I- 28366 cubic feet) i bushel 2 bushels I strike. 4bushels i coomb. 5 bushels I sack. 8 bushels i quarter. 4 quarters (41.Q77 cubic feet) i chaldron. 5quarters i wey or load. 2 loads I last The standard bushel is 18^ inches in diameter inside, and 8^ inches deep; it holds 80 pounds of distilled water at 62'' F. It is 19^^ inches in diameter outside. This measure is applied to dry goods, as com, seeds, roots, &C., which are struck with a straight bar. The old dry measures had the same denominations and proportions, and were 96.95 per cent of the Imperial Dry Measures, above given. The heaped imperial bushel must be an upright cylinder, of which the diameter is not less than twice its depth, and the height of the conical heap must be at least three-fourths of the depth of the bushel, the outside of the pleasure being the boundary of the base of the cone. It may be 18.789 inches in diameter inside, and 8 inches deep; and the capacity, heaped, must be 1.6293 cubic feet. Heaped measure is used for such goods as camiot be conveniently stricken, as coals, potatoes, and fruit Coal Measure, 3 bushels (heaped) i sack. 9 bushels I vat 36 bushels, or 1 2 sacks i chaldron. 5^ chaldrons i room. 21 chaldrons i score. Old Wine and Spirit Measure, •11 \ Imperial 4 g^lS I pint Gafions. 2 pints I quart 4 quarts (231 cubic inches) i gallon = .8333 10 ^lons ranker = 8.333 18 gallons I runlet = 15. 31}^ gallons I barrel = 26.250 42 gallons I tierce = 35. 'jgS-"'} hogshead =5.5 .^*CS^°J} , p»,cheo„ . 70. ; 126 gallons, or \ 2 hogsheads, or > i pipeorbutt= 105. i^ puncheons ) 2 pipes, or ) ^ .^ -^,r. *^ '^ i > I tun =210. 3 puncheons j I40 WEIGHTS AND MEASURES. By this measure wines, spirits, cider, perry, mead, vinegar, oil, &c., are measured; but the contents of every cask are reckoned in imperial gallons when sold. The imperial gallon is one-fifth laiger than the old wine gallon. Old Ak and Beer Measure, 2 pints I quart JSESS!* 4 quarts (282 cubic inches) i gallon = 1.017 9 gallons I firkin = 9.153 2 firkins, or 18 gallons 1 kilderkin = 18.306 2 kilderkins, or I , u^^^i r a , 11 ' > I barrel = 36.612 36 gallons J ^ zoiX'"} ^''"" =^°^-«^^ The imperial gallon is one-sixtieth smaller than the old beer gallon. Apothecaries Fluid Measure, 60 minims (ni) i fluid drachm (/5). 8 drachms (water, 1.732- cubic) « • , / ^ ^x inches, is? >^ grains) } ^ ^^'^ ^"""^ (/ D- 20 ounces i pint ( ^ )• 8 pints (water, 70,000 grains) i gallon {i^^-)- 1 drop I grain. 60 drops I drachm. 4drachms i tablespoonfiiL 2 ounces (water, 875 grains) i wineglassful. 3 ounces i teacupful. V. Measures of Weight. — Tables No. 26. Avoirdupois Weight, 16 drachms, or I / v ,^^1/ ,^:„« \ I ounce(^2:.). 437?^ grams j • ^ ' 16 ounces, or ) j /• • i\ //z \ 7000 grains } ^ P^""*' (""Penal) {ib.). 8 pounds I stone (London meat market). 14 pounds I stone. 28 pounds, or I ^ / \ 2 stones I ' quarter (^r.). 4 quarters, or \ 8 stones, or > i hundredweight {cwt). 112 pounds j 20 hundredweights i ton. The grain above noted, of which there are 7000 to the pound avoirdupois, is the same as the troy grain, of which there are 5760 to the troy pound. Hence the troy pound is to the avoirdupois pound as i to 1.2 15, or as 14 to 17. \ GREAT BRITAIN AND IRELAND.— WEIGHTS. I41 The troy ounce is to the avoirdupois ounce as 480 grains, the weight of the former, to 43 7 >^ grains, the weight of the latter; or, as 1 to .9115. In Wales, the iron ton is 20 cwt of 120 lbs. each. Troy Weight. 24 grams i pennyweight (^/o^/.). 20 pennyweights, or ) 480 grains / ' ^^"^^• 1 2 ounces, or ) • 5760 grains / ^ P°"°<^ 25 pounds I quarter. 4 quarters, or loo poUnds i hundredweight. By troy weight are weighed gold, silver, jewels, and such liquors as are sold by weight Diamond Weight I diamond grain 0.8 troy grain. I carat 4 diamond grains. 15J4 carats i troy ounce. Apothecaries Weight. The revised table of weights of the British Pharmacopeia is as follows : it is according to the avoirdupois scale : — 4375^ grains.... i ounce. 16 ounces i pound. In the old table of Apothecaries* Weight, superseded by the preceding table, the troy scale was followed, thus: — Old Apothecaries Weight. 20 grains i scruple O). 3 scruples, or ) j 1. /-\ 6ograi^ I I drachm (3). 8 drachms, or ) ' /«\ 480 grains } ^ °"'»^« <?)• 12 ounces, or ) j /7il\ 5760 grains } • ^ POund (/*.)• Weights of Current Coins. I farthing, .8 inch diameter, »/,o ounce. I halfpenny, i.o „ Yj „ I penny, 1.2 „ V3 » I threepenny piece »/«> ^i I fourpenny piece '/xs „ I sixpence Yio „ I shilling V5 „ I florin »/s » I half-crown Ya „ 5 shillings or 10 sixpences i „ I sovereign »/^ ounce, nearly. For the exact weight in grains of these coins, see Table of British Money. 142 WEIGHTS AND MEASURES. Coal Weight. 14 pounds I stone. 28 pounds I quarter hundrpdweighL 56 pounds I half hundredweight. 88 pounds i bushel.* I sack, of 1 1 2 pounds i hundredweight I double sack, of 224 pounds... 2 hundredweights. 20 hundredweights, or I , 10 double sacks J 26j^ hundredweights i chaldron (London). 53 hundredweights i chaldron ( Newcasde). 7 tons T room. 21 tons 4 cwt I barge or keel. * Sundry Bushel Measures, I Cornish bushel of coal is 90 or 94 pounds ;. heaped, loi pounds. I Welsh bushel, average wei£[ht 93 pounds. I Newcastle bushel is 80 or 84 pounids. Bradley Main, 92^ pounds. I London bushel, 80 or 84 pounds. f In Wales the miners* coal-ton is 21 cwt. of 120 lbs. each. Wool Weight ' 7 pounds I clove. 2 cloves, or 14 pounds i stone. 2 stones I tod. 6]4 tods I wey. 2 weys I sack. 12 sacks, or 39 hundredweight i last. 12 score, or 240 pounds i pack. Hay and Straw Weig/U. I truss of straw 36 pounds. I load of straw 11 hundredweights, 64 pounds. I truss of old hay 56 pounds. I load of old hay 18 hundredweight. I cubic yard of old hay 9 stone. I cubic yard of oldish hay 8 stone. I truss of new hay 60 pounds. 1 load of new hay ; 19 hundredweights, 3 2 pounds. I cubic yard of new hay 6 stone. Com and Flour Weight 1 peck, or stone of flour 14 pounds. 10 pecks I boll = 140 2 bolls I sack =280 14 pecks I barrel =196 I bushel of wheat 60 I bushel of barley ^ 47 I bushel of oats 40 Six bushels of wheat should yield one sack of flour; i last of corn is 80 bushel& GREAT BRITAIN AND IRELAND.— MISCELLANEOUS. I43 Miscellaneous Tables. — No. 27. Whatman* s Drawing Papers, — Sizes of Sheets, Antiquarian 53 inches long, 3 1 inches wide. Double-elephant 40 Atlas 34 Colombier 34 Imperial 30 Elephant 28 Super-royal 27 Royal 23 Medium 22 Demy 20 n 27 26 23 22 23 19 19 17 15 Commercial Numbers and Stationery. 12 articles 1 dozen. 13 articles i long dozen. 12 dozen i gross. 20 articles i score. 5 score I common hundred. 6 score i great hundred. 30 deals I quarter. 4 quarters i hundred. 24 sheets of paper i quire. 20 quires i ream. 2ij| quires i printers* ream. 5 dozen skins of parchment i roll. Measures relating to Building, Load of timber, unhewn or rough 40 cubic feet Load, hewn or squared [ 5° cubic feet, reckoned * ^ (to weigh 20 cwt Stack of wood io8 cubic feet. Cord of wood 128 „ (In dockyards, 40 cubic feet of hewn timber are reckoned to weigh 20 cwt. ; 50 cubic feet is a load.) 100 superficial feet i square. Himdred of deals 120 deals. Load of i-inch plank 600 square feet. (Load of plank more than i-inch thick = 600 -^ thickness in inches. Planks, section 11 by 3 inches. Deals, section 9 by 3 „ Battens, section 7 by 2^ „ A reduced deal is 1 J^ inches thick, 1 1 inches wide, and 1 2 feet long. Bundle of 4 feet oak-heart laths 120 laths. Load of „ „ 3 7 J^ bundles. Bundle of 5 feet oak-heart laths 100 laths. Load of „ „ 30 bundles. 144 WEIGHTS AND MEASURES. Measures reiaiing to Building {continued,) Load of statute bricks 500. Load of plain tiles 1000, Load of lime 32 bushels. Load of sand 36 „ Hundred of lime 35 „ Hundred of nails, or tacks 120. Thousand of nails, or tacks 1 200. Fodder of lead iqJ^ cwt. Sheet lead 6 to 10 pounds per sq, ft Hundred of lead 112 pounds. Table of glass 5 feet. Case of glass 45 tables. case of glass { ^^^"^^f taS~'' Stone of glass 5 pounds. Seam of glass 24 stone. Sundry Commercial Measures, Dicker of hides ; 10 skins. Last of hides 20 dickers. Weigh of cheese 256 pounds. Barrel of herrings 26 V3 gallons. Cran of herrings 37^ „ Pocket of hops i ^ to 2 cwt. Bag of hops 3j^ cwt, nearly. Last of potash, cod-fish, white her- ) barrels. rings, meal, pitch, tar j Barrel of tar 26 J4 gallons. Barrel of anchovies 30 pounds. Barrel of butter 224 „ Barrel of candles 120 „ Barrel of turpentine 2 to 2ji c^vt. "Barrel of gunpowder 100 pounds. Last of gunpowder 24 barrels. Measures for Ships, I ton, displacement of a ship, 35 cubic feet I ton, registered internal capacity of do., 1 00 do. I ton, shipbuilders' old measurement, 94 do. Comparison of Compound Units. — ^Tables No. 28. Measures of Velocity, ^-y ^^ji. S ^•467 feet per second. I mile per hour { 88.0 feet per minute. I knot per hour i,688 feet per second. I foot per second .682 mile per hour. I foot per minute .01136 mile per hour. GREAT BRITAIN AND IRELAND. — COMPOUND UNITS. 14$ Measura of Volume and Time, I cubic foot per second [ ^""^ ^"^!^ ^^^^ P^^' T'"'*^- I ^33-333 cubic yards per hour. I cubic foot per minute 2.222 cubic yards per. hour. I cubic yard per hour .45 cubic foot per minute. I cubic inch per second [ ''•^^ cubic foot per hour. ^ ( 12.984 gallons per hour. I gaUon per second 569. 124 cubic feet per hour. I gallon per min ute 9-485 cubic feet per .hour. Measures of Pressure and Weight (See also page 127.) i 144 lbs. per square foot. I lb. per square inch < 1296 lbs. per square yard. i -57^6 ton per square yar^. 1 atmosphere (14.7 lbs.) per ) g ^^^ ^^ square mch j ^ ^ v ^ j i .00694 lb. per square inch. I lb. per square foot < .11 ii ounce per square inch. ( .0804 cwt per square yard. ,, . , ( 2.0355 inches of mercury at 32° F. 1 lb. per square inch -^ ^.j^^^f^et of water at s l\z F. . , X. 1. o -c f .401 lb. per square inch. I inch of mercury at 32 F. -j ^ ^^^ feet of water at S2".3 F. ( -4333 lb. per square inch. I foot of water, at S2°.3 F. .. <| 62.4 lbs. per square foot. ( .8823 inch of mercury at 32® F. Measures of Weight and Volume, {405. 1 grains per cubic inch. .926 ounce per cubic inch. 4.107 cwt. per cubic yard. 1.205 tons per cubic yard. , . . , f ^•O'^o ounces per cubic foot. 1 grain per cubic mch { ^ ^J^ pounds per cubic foot I ounce per cubic inch 1 08 pounds per cubic foot. 1 cwt. per cubic yard 4. 1 48 pounds per cubic foot. I ton per cubic yard 8 2. 963 pounds per cubic foot r I pound for 1 1 2 2 cubic feet. I grain per gallon (i in 70,000 parts by weight, of water) I pound for 41.5 cubic yards. I pound for 31.8 cubic metres. 220 grains for i cubic metre. .503 ounce for i cubic metre. Measures of Power, w. ce \ xj T> ( 1,980,000 foot-pounds per lb. of fuel. 1 lb. of tuei per n.i'. \ ^21.76 million foot-pounds per cwt of fuel P^^ ^^"^ i 2,565 units of heat i,ooo,ooofoo^p^^^^ I j^3 p^^^^g ^^f^^i p^^ H.P. per hour. 146 WEIGHTS AND MEASURES. FRANCE.— THE METRIC STANDARDS OF WEIGHTS AND MEASURES. The primary metric standards are : — the metre, the unit of length ; and the kilogramme, the unit of weight, derived from the metre : being the two platinum standards deposited at the Palais des Archives at Paris. The standard metre is defined to be equal to one ten-millionth part of the quadrant of the terrestrial meridian, that is to say, the distance from the equator to the pole, passing through Paris, which, by the latest and most authoritative measurement, is 39.3762 inches, in terms of the Imperial standard at 62° F. By the latest and most accurate measurement, the actual standard metre at 32° F. is, in terms of the Imperial standard at 62° F., 39.37043 inches; and its legal equivalent, declared in the Metric Act of 1864, is ^9.3708 inches, being the same as that adopted in France. The standard kilogramme (looo grammes) is defined to be the weight of a cubic decimetre of distilled water at its maximum density, at 4^0 C. or 39°. I F. This is legally taken to be .2.20462125 lbs., or , 2 lbs., 3 oz., 4.383 drachms, or i5»432.34874 grains. There is in the Standard Department at Westminster a newly-constructed subdivided standard yard, laid down upon a bar of Baily's metal, upon which a subdivided metre has also been laid down. The metric unit of capacity is the litre, defined to be equal to a cubic decimetre. Its Imperial equivalent is 0.22009 gallon. There is no other official standard of weight and measure in France than the metre and the kilogramme; there is no standard litre or unit of capacity. The metric system is not really founded on the length of a quadrant of the meridian, and although it is described as a scientific system, because of the simple and definite relation between the metre, which is its basis and unit of length, and the kilogramme and litre, which are the units of weight and capacity, it is admitted that it has been found impossible practically to carry it out with scientific accuracy. The standard kilogramme is admitted not to be actually the weight of a cubic decimetre of pure water at the specified temperature, nor the litre a measure of capacity holding a cubic decimetre of pure water. The real standard unit of weight is declared, even by men of science in France, to be merely the platinum kilogramme-weight deposited at the Palais des Archives, as the real standard unit and basis of the metric system is the platinum metre, also deposited there. It is an accomplished fact, however, that all civilized nations have tacitly agreed to recognize the metric system as affording for the future the advantages of a universal system of weights and measures, and to adopt the standards deposited at the Palais des Archives as the primary units of the system. The French metric system has been adopted, and its use made compul- sory by the following States: — France and Belgium, in 1801; Holland, in 1819; Greece, in 1836; Italy and Spain, in 1859; Portugal, in 1860--68; the German Empire, in 1872; Colombia, Venezuela, in 1872; Ecuador, FRANCE.— THE METRIC STANDARDS — LENGTH. I47 Bcazfl, Peru, and Chili, in i860; also by the Argentine Confederation, and Uruguay. Great Britain and Ireland, in 1864, adopted the metric system, so far as to render contracts in terms of the French metric S3rstem permissive. The United States of North America, in 1866, legalized the French metric system concurrently with the old system; it was also legalized in British North America. Switzerland, in 1856, legalized the foot of three decimetres as the unit of length, with a decimal scale; the unit of weight being the pound of 500 grammes, or half a kilogramme, with two distinct scales of multiples and parts, one decimal, the other according to the old custom. Sweden, in 1855, by a law made compulsory in 1858, adopted a decimal system of weights and measures, having for the unit of length a foot of 0.297 metre, and the unit of weight a pound of 0.42 kilogramme: — ^being the original units decimally treated. Denmark adopted the metric system so far as the pound of 500 grammes. The pound is decimally treated, and since 1863 the use of the greatest parts of the multiples of the pound not conformable to decimal sub- division has been prohibited. Austria, in 1853, adopted a pound of 500 grammes, with decimal divisions, for customs and ^scaJ purposes. Russia awaits the example of those countries with which she has conunercial relations, especially of England. In Morocco and Tunis, the weights and measures have no relation with the metric system. On the 20th May, 1875, the international convention for the adoption of the French metrical system of weights and measures was signed at Paris by the plenipotentiaries of France, Austria, Germany, Italy, Russia, Spain, Portugal, Turkey, Switzerland, Belgium, Sweden, Denmark, the United States, the Argentine Republic, Peru, and Brazil. A special clause reserves to States not included in the above list the right of eventually adhering to the convention. I. French Measures of Length. — Table No. 29. I millimetre 10 centimetres. 10 centimetres i decimetre. 10 decimetres, or ] 100 centimetres, or > i metre. 1000 millimetres j 10 metres i decametre. 10 decametres x i hectometre. 10 hectometres, or 1000 metres i kilometre {kilo,) 10 kilometres i myriametre. I toise (old measure) =1.949 metres. 1000 toises I mille = 1.949 kilometres. 2000 toises I itinerary league =3.898 „ 2280.329 toises I terrestrial league =4.444 „ 2850.411 toises I nautical league =5-555 n X noeud (British nautical mile) = 1.855 „ 148 WEIGHTS AND MEASURES. French Wire-Gauges {Jauges de Fils de Fer), The French wire-gauge, like the English, has been subject to variation. Table No. 30 contains the values of the "points," or numbers, of the Limoges gauge; table No. 31 gives the values of a wire-gauge used in the manufacture of galvanized iron; and table No. 32 the values of a gauge which comprises wire and bars up to a decimetre in diameter. French Wire-Gauge i [J<^uge de Limoges) .—Table No. 30. Number. Diameter. Number. Diameter. • Number. Diameter. MUUmetreJ Inch. Millimetre. Inch- Millimetre. Inch. •39 .0154 9 1.35 .0532 18 3- 40 .134 I .45 .0177 10 1.46 •0575 19 3-95 .156 2 .56 .0221 II 1.68 .0661 20 4-50 .177 3 .67 .0264 12 1.80 .0706 21 510 .201 4 .79 .0311 13 1.91 .0752 22 5-65 .222 5' .90 •0354 14 2.02 •0795 23 6.20 .244 6 1. 01 .0398 15 2.14 .0843 24 6.80 .268 7 1. 12 .0441 16 2.25 .0886 8 1.24 .0488 1 17 2.84 .112 French Wire-Gauge for Galvanized Iron Wire. — ^Table No. 31. Number. Diameter. Number. Diameter. Number. Diameter. M'metre. Inch. M'metre. Inch. M'metre. Inch. I .6 .0236 9 1.4 .0551 17 30 .118 2 .7 .0276 10 1-5 .0591 18 3-4 .134 3 .8 •0315 II 1.6 .0630 19 3.9 .154 4 •9 •0354 12 1.8 .0709 20 4.4 .173 5 I.O •0394 13 2.0 .0787 21 4.9 .193 6 I.I .0433 14 2.2 .0866 22 5-4 .213 7 1.2 .0473 15 2.4 .0945 23 5-9 .232 8 1.3 .0512 16 2.7 .106 French Wire-Gauge. — Table No. 32. Mark. Size. Mark. Size. Mark. Size. Mark. Size. Millimetre. Millimetre. Millimetre. Millimetre. P 5 8 13 16 27 24 64 I 6 9 14 17 30 25 70 2 7 10 IS 18 34 . 26 76 3 8 II 16 19 39 27 83 4 9 12 18 20 44 28 88 5 10 13 20 21 49 29 94 6 II 14 22 22 54 30 100 7 12 15 24 23 59 FRANCE. — THE METRIC STANDARDS. I49 II. French Measures of Surface. — Table No. 33. 1 00 square millimetres i square centimetre. 1 00 square centimetres i square decimetre. 100 square decimetres, or ) ^ ^. ^ ' > I square metre, or centiare. r 0,000 square centimetres / ^ ^ '-^ > ^ ^ 100 square metres, or centiares... i square decametre, or are. 100 square decametres, or ares ... i square hectometre, or hectare. 100 square hectometres, or hectares 1 square myriametre. liand is measured in terms of the centiare^ the are, and the hectare or arpent metrique {metric acre). There is also the decare, of 10 ares. III. French Measures of Volume. — Tables No. 34. Cubic Measure. 1 000 cubic millimetres i cubic decimetre. 1000 cubic decimetres i cubic metre. Wood Measure, 10 decist^res i stfere* (i cubic metre). I voie (Paris) 2 stferes. I voie de charbon (charcoal) 0.2 stfere ( 75 cubic metre). I corde 4 stferes. * The stire measures 1. 14 metres x 0.88 metre x i metre, the billets of wood being 1. 14 metre in length. IV. French Measures of Capacity. — ^Tables No. 35. Liquid Measure, 10 centilitres i decilitre. 10 d^cihtres i litre. 10 litres I decalitre. Dry Measure. 10 litres I decalitre. 10 decalitres, or) ^ hectoUtre. 100 litres J 10^ liS"*'^' °'} ' '^""^^ <' ^^'*= ™*''^>- The use of measures equal to a double-litre^ a half-litre, a double-dicilitre^ a half-d£cilitre^ is sanctioned by law. ISO WEIGHTS AND MEASURES. V. French Measures of Weight. — Table No. 36. 10 milligrammes i centigramme. 10 centigrammes i decigramme. 10 decigrammes i gramme. 10 grammes i decagramme. 10 decagrammes i hectogramme. 10 hectogrammes, or ) ^ kilogramme (M., kUc^.) 1000 grammes J \ > c / 10 kilogrammes i myriagramme. 10 myriagtammes, or ) ^ j^ ^ ^^ 100 kilogrammes J ^ ^ 10 quintaux, or ) ( i millier, tonneau de mer, or tonne 1 000 kilogrammes j ( (weight of i cubic metre of water at 39°. i ). EQUIVALENTS OF BRITISH IMPERIAL AND FRENCH METRIC WEIGHTS AND MEASURES. I. Measures of Length. — ^Tables No. 37. A DBCIMBTRB DIVIDBD INTO CBNTIMBTRBS AND MILLIMBTRBS. 2 a llljl.nlllllllHIllllliJhllllini.ll.limlhTT llllllill JTTTT 'mill. I '"iImii iiimm UL It I I I I I I I I 11 M |l IIMII erg: MM ' ' ' ' I I I .1 i! INCHBS AND TBNTHS. Mbtsic Dbnominations AND VaLUBS. • Equivalbnts in Impbrial Dbnominations. Metres. Inches. Feet. Yards. « MUes. I millimetre I centimetre I decimetre I M£1K£ «•.. I dekametre I hectometre I KILOMETRE I myriametre /xooo /lOO V.O I 10 100 1,000 10,000 = 0-03937 = 0.39370 = 3-93704 = 39-37043 = 3.28087 = 32.80869 = 3280.87 = 1.09362 10.93623 109.36231 = 1,093.6231 = 10,936.231 = 0.62138 = 6.21377 IMPERIAL AND METRIC EQUIVALENTS. IS! Tables No. 37 {continued). Impbrial Denominations. Equivalents in Metric Denominations. Centunetres. Metres. Kilometres. 1 I inch I2K.A millimetres) = 2.5399s = , 0.30480 0.91439 1.82878 5.02915 = 20.11662 = 201.1662 = 1,609.3296 = 0.20117 = 1.60933 I foot, or 12 inches I yard, or 3 feet, or 36 inches.... I fathom, or 2 yards, or 6 feet.... I pole, OT K'^A yards I chain, or 4 poles, or 22 yards... I furlong, 40 poles, or 220 yards I mile, 8 furlongs, or 1760 yards Equivalent Values of Millimetres and Inches. — ^Tables No. 38. Millimetres = Inches. MiOimetRs. Inches. MUiimetres. Inches. Millimetres. Inches. Millimetres. Inches. I .0394 27 1.0630 53 2.0866 79 3-"03 2 .0787 28 1. 1024 54 2.1260 80 3.1496 3 .1181 29 I.I417 55 2.1654 81 3.1890 , 4 -1575 3<5 I.181I 56 2.2047 82 3.2284 5 .1968 31 1.2205 57 2.2441 83 3.2677 6 .2362 32 1.2598 58 2.2835 84 3-3071 7 .2756 33 1.2992 59 2.3228 85 3-3465 8 .3150 34 1.3386 60 2.3622 86 3-3859 9 •3543 35 1.3780 61 2.4016 87 3-4252 10 .3937 36 1-4173 62 2.4410 88 3.4646 II .4331 37 1.4567 63 2.4803 89 3-5040 12 .4724 38 1.496 1 64 2.5197 90 3-5433 13 .5118 39 1-5354 65 2.5591 91 3-5827 14 .5512 40 1.5748 66 2.5984 92 3.6221 ; '5 .5906 41 1. 6142 67 2.6378 93 3.6614 16 .6299 42 1.6536 68 2.6772 94 3.7008 17 .6693 43 1.6929 69 2.7166 95 3-7402 18 .7087 44 1-7323 70 2.7559 96 3.7796 19 .7480 45 1.7717 71 2.7953 97 3.8189 20 .7874 46 1.8110 72 2.8347 98 3.8583 1 21 .8268 47 1.8504 73 2.8740 99 3-8977 '- 22 ,8661 48 1.8898 •74 2.9134 100 3-9370 23 .9055 49 1. 9291 75 2.9528 = 1 de cimetre. 24 .9449 50 1.9685 76 2.9922 ' 25 .9S43 51 2.0079 77 3-0315 i ^' 1.0236 52 2.0473 78 3.0709 152 WEIGHTS AND MEASURES. Tables No. 38 {continued), InCHBS DbCIMALLY = MiLLIMBTRBS. Inches. 1 Millimetres, i Inches. Millimetres. Inches. Millimetres. Inches. Millimetres. 239 .01 -5 1 .26 6.60 .60 152 .94 .02 •51 .28 7.II .62 15-7 .96 24.4 •03 .76 •30 7.62 .64 16.3 .98 24-9 .04 1.02 .32 8.13 .66 16.8 1. 00 25.4 •05 1.27 •34 8.64 .68 17-3 2.00 50.8 .06 1-52 .36 9.14 .70 17.8 3.00 76.2 .07 1.78 .38 9^65 .72 18.3 4.00 IOI.6 .08 2.03 .40 10.2 .74 18.8 5.00 127.0 .09 2.29 .42 10.7 .76 19-3 6.00 152.4 .10 2.54 .44 II. 2 .78 19.8 7.00 177.8 .12 3.05 .46 II.7 .80 20.3 8.00 203.2 .14 3.56 .48 12.2 .82 20.8 9.00 228.6 .16 4.06 •50 12.7 .84 21.3 10.00 254.0 .18 4.57 •52 13.2 .86 21.8 11.00 279.4 .20 5.08 •54 13-7 .88 22.4 12.00 304.8 .22 5-59 .56 14.2 .90 22.9 = I foot. .24 6.10 .58 14.7 •92 23.4 Inches IN Fractions = Millimetrks. Eighths. Sixteenths. Thirty-seconds. Millimetres. Eighths. Sixteenths. Thirty-seconds. Millimetres. I •79 17 135 I 2 1-59 9 18 143 3 2.38 19 I5-I I 2 4 3-17 5 10 20 159 5 3-97 21 16.7 3 6 4.76 II 22 175 7 5-56 23 18.3 2 4 8 6.35 6 12 24 19.0 9' 7.14 25 19.8 5 10 7^94 13 26 20.6 II 8.73 27 21.4 3 6 12 9^52 7 14 28 22.2 13 10.32 29 23.0 7 14 II. II 15 30 23.8 15 II. 91 31 24.6 4 8 16 12.7 8 16 32 25.4 By means of the preceding tables of equivalent values of inches and millimetres, the equivalent values of inches in centimetres and decimetres, and even in metres, may be found by simply altering the position of the decimal point. This method naturally follows from the decimal subdivisions of French measure. Take, for example, the tabular value of i millimetre, and shift the IMPERIAL AND METRIC EQUIVALENTS. 153 decimal pK>int successively, by one digit, towards the right-hand side; the values of a centimetre, a decimetre, and a metre are thereby expressed in inches, as follows: — I millimetre 0394 inches. I centimetre o-394 I decimetre 3.94 I metre 39.4 At the same time, it appears that, by selecting the tabular value of 10 millimetres, the value of its multiples are given more accurately, thus, — 10 millimetres, or i centimetre 0-3937 inches. I decimetre 3-937 »» I metre 39.37 „ Again: — 100 millimetres, or i decimeti^ = 3-937 inches. I metre =39-37 n Similarly, for example : — .32 inch = 8.13 millimetres. 3.2 „ = 81.3 „ ^ f 813.0 „ or ( .813 metre. 32.0 » II. Square Measures, or Measures of Surface. — Tables No. 39. Mbtric I square centimetre 1 square decimetre I square metre, or centiare I ARE, or square dekametre, or 100 square metres 1 hectare, or metrical acre, or 100 ares, or 10,000 square metres Imperial Square Measures. .155 square inch. 15.5003 square inches. 10.7641 square feet, or 1. 1960 square yards. 1076.41 square feet, or 119.60 square yards. 1,960.11 square yards, or 2.4711 acres, or acres and 2280.1240 square yards. Imperial = Metric Square Measures. I Imperial Measures. S<}uare Centimetres. Square Metres. Ares. Hectares. f snuare incli r = 6.45148 = 0.092901 = 0.836 II 2 = 25.292 = 1011.696 = 4046.782 = 10.11696 = 40.4678 = 0.40468 = 258.98944 1 square ft., or 144 sq. inches I square yard, or 9 square ) tcet, or 1296 sq. inches ) I perch or rod, or 30X | square yards y I rood, or 40 perches, or") I 1 2 10 square yards ) I acre, or 4 roods, or 4840 ) square yards J I square mile, or 640 acres 154 WEIGHTS AND MEASURES. III. Cubic Measures. — Tables No. 40. Metric = Imperial Cubic Measures. I cubic centimetre = 0.061025 cubic inch. u* J -^ 4- (61.02524 cubic inches, or I cubic deametre = | 0.0353156 cubic foot , . _, , f 35.^156 cubic feet, or I cubic metre = { 1.308 cubic yanis. Imperial = Metric Cubic Measures. 1 cubic inch = 16.387 cubic centimetres. I cubic foot I cubic yard _ 128.31531 cubic decimetres, or ~ ( 0.02I 1283 1 6 cubic metre. 0.76453 cubic metre. Wood Measure. «'«. " »"« »"" { ^fisfcS^jSa I decistfere * 3.5316 cubic feet. I voie de bois (wood), or 2 stores. Paris { '^it'.l^^^y^^l I voie de charbon (charcoal) = i sack ( S}^ bushels, or = */5 stfere ( 7.063 cubic feet I corde of wood = 4 cubic metres 141.26 cubic feet IV. Measures of Capacity. — Tables No. 41. or Metric Dekominations AND Values. Equivalents in Imperial Denominations. Litres. Gills. Pints. Quarts. Gallons. Bushels. a Quarters. Centilitre Decilitre Litre (61.02524c. in.) Dekalitre Hectolitre Kilolitre /lOO }' 10 100 1000 0.0704 0. 7043 0.0176 O.I761 1.7607 0.8804 0.2201 2.2009 22.009 220.09 0.2751 2.75II 27.511 0.344 3.439 Imperial Denominations. Equivalents in Metric Denominations. Litres. Dekalitres. Hectolitres. I gill I pint, or 4 gills I quart, or 2 pints I gallon, or 4 Quarts = 0.1420 = 0.5679 = I.I359 = 4.5436 = 9.0872 = 36.3488 = 290.7904 = 0.9087 = 3.6349 = 29.0790 = 2.9079 I peck, or 2 crallons I bushel, or 8 gallons I Quarter, or 8 bushels IMPERIAL AND METRIC EQUIVALENTS. ISS V. Measures of Weight. — Tables No. 42. MsTKic Weights = Impbkial AvoiitDUPOis Weights. I kilogramme = 2 lbs. 3 oz. 4 drachms^ 10.47374 grains. Mbthic Weights. Equivalbnts im Imperial Denominations. Grammes. Grains. Ounces. 1 Pounds. Hundred- weights. Tons. M illigramme Vxooo Vioo I 10 100 1,000 10,000 100,000 1,000,000 0.01 54 0.1543 1.5432 154323 154.3235 1543.2349 15432.3487 0.3527 3.5274 35.2739 2.2046 22.0462 220.4621 2204.6212 1.9684 19.6841 1 II ! lit CentigrazDzne Decieramme Gramme Dekagramme Hectogramme Kilogramme Myriagramme Quintal^ or 100 kilog. Millier, or metric ton Imperial Avoirdupois = Metric Weights. iMPBfiiAi. Avoirdupois Weights. Grammes. Decigrammes. Kilogrammes. Millier, or Metric Ton. I diachni = I.77184 - 28.34954 = 453.59265 = 2.83495 = 45.35926 0.45359 == 50.80237 = IO16.O4754 = 1. 01604 I ounce, or 16 drams I pound, or 1 6 ounces I hundredweight, ) or 112 pounds ) I ton, or 20 hun- \ dredweights j Metric Weights = Imperial Troy Weights. I kilogramme = 2 troy lbs. 8 oz. 3 dwts., .34874 grain. Hetkxc Weights. Grains. Pennyweights. Ounces. Troy Pound. 1 Milligramme... 0.01543 ^-i Centigramme ... 0.15432 Deagramnie . . . 1.54323 GRABHtfE. 15-43234 Dekagramme... = '154.32349 = 0.64301 = 0.32151 Hectogramme.. = 1543.23487 = 6.43014 = 3-21507 Kilogramme... = I51432.34874 = 32.15073 = 2.67922 156 WEIGHTS AND MEASURES. Imprxial Troy = Metric Weights. Imperial Troy Weights. Equivalents in Metric Denominations. Millignuiime. Gramme. Dekagramme. Hecto- gramme. Kilo- gramme. I troy grain I „ dwt, or 24 gr. I „ oz., or 480 „ I „ lb., or 5,760 „ 64.79895 0.06480 1-55517 31.10349 373-24195 3.IIO35 37.32419 373242 0-37324 APPROXIMATE EQUIVALENTS OP ENGLISH AND FRENCH MEASURES. The following are approximately equal English and French measures of length : — I pole, or perch {s}4 yards)... 5 metres (exactly 5.029 metres). I chain (22 yards) 20 metres (exactly 20. 1 1 66 metres). I furlong (220 yards) 200 metres (exactly 201.166 metres). 5 furlongs I kilometre (exactly i .0058 kilometres). ^ . (3 decimetres (exactly 3.048 decimetres), or ^ ^ \ 30 centimetres. One metre = 3.28 feet = 3 feet 3 inches and 3 eighths all but Vsxa inch; = 40 inches nearly ( ^/e^th. or 1.6 per cent less). .100 metre (i decimetre) .010 metre (i centimetre) , , __ , , ^. .001 metre (i millimetre) = .04 inch, or Viooths inch, or two-thirds of Vx6 inch, or 725 inch, nearly. One inch is about 2}4 centimetres (exactly 2.54). One inch is about 25 millimetres (exactly 25.4). One yard is "/"ths of a metre. 11 metres are equal to 12 yards. Approximate rule for converting metres, or parts of metres, into yards : — Add Vxi^h {}( per cent. less). For converting metres into inches: — Multiply by 40; and to convert inches into metres, or parts of metres, divide by 40. One kilometre is about ^ mile (it is 0.6 per cent. less). One mile is about 1.6 or 1 3/^ kilometres (it is 0.6 per cent less) == 16 10 metres, about. « With respect to superficial measures : — One square centimetre is about 7*6.5 part of a square inch. One square inch is equal to about 6.5 square centimetres. One square metrecontains fully 10 J^ square feet, or nearly i^j square 3rards. One square yard is nearly ^7 ths of a square metre. One acre is over 4000 square metres (about 1.2 per cent more). One square mile is nearly 260 hectares (about 0.4 per cent less). FRENCH AND ENGLISH COMPOUND UNITS. 1 57 With respect to cubic measures, and to capacity : — One cubic yard is about 5^ cubic metre (it is 2 per cent. more). One cubic metre is nearly i^ cubic yard (it is i^ per cent. less). One cubic metre is nearly 35 'A cubic feet (it is .05 per cent. less). One litre is over i^ pints (it is 0.57 per cent more). One gallon contains above 4^ litres (it holds about i per cent. more). One kilolitre (a cubic metre) holds nearly i ton of water at 62*^ F. (i^ per cent less). — One cubic foot contams 28.3 litres. With respect to weights: — The ton and the gramme stand at nearly equal distances above and below the kilogramme, thus : — I ton is 1,016,047.5 grammes, I kilogramme is 1,000.0 grammes, I gramme i.o gramme, in the ratio of about 1,000,000 : 1,000 : i. One gramme is nearly 15^ grains (about yi per cent. less). One kilogramme is about 2 '/s pounds avoirdupois (about V4 per cent, more). A thousand kilogrammes, or a metric ton, is nearly one English ton (about I J^ per cent less). One hundredweight is nearly 5 1 kilogrammes ( 2/5 per cent less). EQUIVALENTS OF FRENCH AND ENGLISH COMPOUND UNITS OF MEASUREMENT. Weighty Pressure^ and Measure, • kiX*"-- P« "■""••■ { ;IM JSjl'^Taii I pound per foot 1.488 kilogrammes per metre. 1 pound i>er yard .496 kilogramme per metre. 1000 kilc^ammes per metre .300 ton per foot I ton per foot 3333-333 kilogrammes per metre. 1000 kfl^mmes, or X tonne, per ) ^^g^ ^^^ ^^ ^^^ I tcm per mile 631.0 kilogrammes per kilometre. , T .„../ 1422.32 pounds per square inch. I kilogramme per square millimetre \ l^^\^^^ ^^^ ^^^^^^^ ^^^^ 1000 Tiounds ner souare inch i -703077 kilogramme per square 1000 pounas per square men | millimetre. , ton per square inch { '"575 ^"^JSTeSe.^" ^' . I kilogramme per square centimetre 14.2232 pounds per square inch. 1-0335 kilogrammes per square centi- ) nounds oer souare inch metre (i atmosphere) / ^^'^ pounds per square men. I pound per square inch I pound per square foot .0703077 kilogramme per square centimetre. 4.883 kilogrammes per square metre. 158 WEIGHTS AND MEASURES. Weighty Pressure^ and Measure {continued), kilogramme per square metre 205 pounds per square foot. centimetre of mercury 394 inch of mercury. inch of mercury 2. 540 centimetres of mercury. centimetre of mercury 193 pound per square inch. pound per square inch 5-i7o centimetres of -mercury. gramme per litre 70.105 grains per gallon. grain per gallon 0143 gramme per litre. kilogramme per cubic metre 0624 pound per cubic foot pound per cubic foot 16.020 kilogrammes per cubic metre. . ^ u- _ * f .984 ton per cubic metre. tonne per cubic metre < \. *, u- ^ ^ ( .752 ton per cubic yard. kilogramme per litre 10.016 pounds per gallon. pound per gallon 998 kilogramme per litre. ton per cubic metre 1.016 tonnes per cubic metre. ton per cubic yard i'329 tonnes per cubic metre. cubic metre per kilogramme 16.020 cubic feet per pound. cubic foot per pound 06 24 cubic metre per kilogramme. ( 1.329 cubic yards per ton. cubic metre per tonne < 1.794 cubic feet per cwt. I 35.882 cubic feet per ton. cubic yard per ton 752 cubic metre per tonne. cubic foot per cwt. . . .^ 557 cubic metre per tonne. cubic foot per ton 0279 cubic metre per tonne. Volumty Area, and Length, cubic metre per lineal metre 1.196 cubic yards per lineal yard. cubic yard per lineal yard ..• 836 cubic metre per lineal metre. cubic metre per square metre 3.281 cubic feet per square foot cubic foot per square foot 3.048 cubic metres per square metre- litre per square metre 0204 gallon per square foot gallon per square foot 48. 905 litres per square metre. i .405 cubic metre per acre, cubic metre per hectare \ .529 cubic yard per acre. ( 89.073 gallons per acre. cubic metre per a^re 2.471 cubic metres per hectare. cubic yard per acre 1.902 cubic metres per hectare. 000 gallons per acre 11.226 cubic metres per hectare. Work, kilogrammetre {ky.m) 7.233 foot-pounds. foot-pound 138 kilogrammetre. cheval-vapeur or cheval (75 -t x « ) horse-power. persecond) / -^ ^ ^ horse-power 1.0139 chevaux. kilogramme per cheval 2.235 pounds per horse-power. pound per horse-power 447 kilogramme per cheval. square metre per cheval 10.913 square feet per horse-power. square foot per horse-power 0916 square metre per cheval. cubic metre per cheval • 35.801 cubic feet per horse-power. cubic foot per horse-power 0279 cubic metre per cheval. FRENCH AND ENGLISH COMPOUND UNITS. 159 Heat. I calorie, or French unit I English heat-unit French mechanical equivalent (424 ) , kilogrammetresj exactly 423.55) ... j "^^ ^' English mechanical equivalent (772 ) foot-pounds) J I calorie per square metre I heat-unit per square foot I calorie per kilogramme 1 heat-unit per pound 10. • 2. I. 968 English heat-units. 252 calorie. 5 foot-pounds. 76 kilogrammetres. 369 heat-unit per square foot. 713 calories per square metre. 800 heat-units per pound. 556 calorie per kilogramme. Speedy 6fc. I I metre per second I kilometre per hour I foot per second, or per minute < ^ \ 1.609 kilometres per hour. , ^.t- _^. J f 35.316 cubic feet per second. I cubic metre per second < ^^^ ^ x^- r ^ • *^ * ( 2119 cubic feet per mmute. I cubic foot per second, or per minute I cubic metre per minute I cubic yard per minute 3.281 feet per second. 196.860 feet per minute. 2.236 miles per hour. .621 mile per hour. .305 metre per second, minute. .447 metre per second. 1.609 kilometres per hoi or per .02 Money, I firanc per kilogramme I penny per pound I shilling per pound I shilling per cwt., or £,1 per ton... < I franc per quintal I franc per tonne < I fianc per metre \ 1 shilling per yard I franc per kilometre < £1 per mile I penny per mile I franc per square metre \ :o cubic teet per secona. cubic feet per minute. 583 cubic metre per second, or per minute. 1.308 cubic yards per minute. .765 cubic metre per minute. 4.320 pence per pound. .360 shilling per pound. 40.320 shillings per cwt., or ^^40.32 per ton. .231 fra^c per kilogramme. 2.772 franc per kilogramme. 24.802 francs per tonne. 2.48 francs per quintal. .403 shilling per cwt. .484 penny per cwt. .806 shilling per ton. .726 shilling per yard. 8.709 pence per yard. 1.378 francs per metre. .0638 J[, per mile. 15.326 pence per mile. 15.660 francs per kilometre. .0652 francs per kilometre. 8.028 pence per square yard. .669 shilling per square yard- l6o WEIGHTS AND MEASURES. I shilling per square yard 1.510 francs per square metre. jCi per square yard 30. 194 francs per square metre. {.270 penny per cubic foot 7.281 pence per cubic yard .607 shilhng per cubic yard. •0303 £ per cubic yard. I penny per cubic foot 3. 708 francs per cubic metre. I penny per cubic yard 137 franc per cubic metre. I shilling per cubic yard i .648 francs per cubic metre. j£i per cubic yard 32.962 francs per cubic metre. I fianc per Utre { 43-270 pence per gallon. ^ \ 3.606 shillmgs per gallon. I franc per hectolitre 1-893 shillings per hogshead (wine). I shilling per hogshead 528 franc per hectolitre. GERMAN EMPIRE. — ^WEIGHTS AND MEASURES. — Tables No. 43. From the ist January, 1872, the French metric system of weights and measures became compulsory throughout the German Empire, as follows : — I. German Measures of Length. French Measure. I Strich = I millimetre, 10 Strichs I New-Zoll = i centimetre. 100 New-Zolls I Stab = i metre. 10 Stabs I Kette = i dekametre. 100 Kettes I Kilometre = i kilometre. 7 Kilometres i Mile - i 7°°^ metres, or 7 Kilometres i Mile - | ^^^ ^^^gj-^j^ ^y^^^ II. German Measures of Surface. I Quadrat-Stab = i square metre. 1 00 Quadrat-Stabs i Ar = 1 00 square metres. . Tj f _ / '^°^ square metres, or '°^^^ I nectar ^i .247 acre. ={'■ III. German Measures of Capacity. I Schoppen = j4 litre. (Beer Measure.) 2 Schoppens i Kanne = i litre. 5° Cannes i Scheffel (bushel) = { ^^.Jj'l^^^erial bushels. ' S^J^effels X Fass (cask) = { J.Jf^^;- The kanne is further divided into measures of j^ kanne, ^ kanne, and ^/x6 kanne. GERMAN EMPIRE. — WEIGHTS, THE FUSS. l6l IV. German Measures of Weight. I Milligramm = i milligramme. lo Milligramms i Centigramm = i centigramme. lo Centigramms i Dezigramm = i decigramme. loo Dezigramms i New-Loth = | '"^ grammes, or I -35273 ounce. ( 500 grammes, or 50 New-Loths........ i Pfund = < ^ kilogramme, or ( I . I o 2 3 pounds avoirdupois. 100 Pfunds I Centner = \ 5° kilogrammes, or \ 110.23 pounds avoirdupois. 2 Centners, or ) Tonne = i ^°^ kilogrammes, or 200 Pfunds / '" ~ I 220.46 pounds avoirdupois. OLD WEIGHTS AND MEASURES OF THE GERMAN STATES. These vary for every state. The chief measures of length are the Fuss, and the EUe, of which the second is in general twice the first. The following are the values of the Fuss, which is the German foot, in the principal states. Values of the German Fuss in the States and Free Towns of THE German Empire. — Table No. 44. Prussia Bavaria Wiirtembeig Saxony Baden Mecklenburg-Schwerin Hesse-Darmstadt Hesse-Cassel Oldenburg Brunswick Hanover Mecklenburg-Strelitz Anhalt Saxe-Coburg-Gotha Saxe-Altenbuig Waldeck Lippe Schwarzburg-Rudolstadt Schwarzburg-Sondershausen : — (i) High Sovereignty and Amstadt ... (2) Low Sovereignty and Sondershausen Reuss Schaumburg-Lippe Hamburg Liibeck Bremen 2.356 inches. 1.491 1.279 1.149 1.811 I.4S7 9-843 1.328 1.649 1.235 1.500 I.4S7 2.356 1.324 1. 122 1. 512 1.398 5-047 1. 149 1.331 1.280 1. 421 1.283 1.324 1.392 11 i9 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 99 l62 WEIGHTS AND MEASURES. KINGDOM OF PRUSSIA.— Old Weights and Measures.— Tables No. 45. I. Prussiai* Measures of Length. English Measure. I Linie = .0858 inch. 12 Linien i ZoU = 1.0297 inches. 12 ZoU I Fuss = j "-356 inches, or I 1.0297 feet. 2 Fuss I EUe = 2.0596 feet 1!?!?°'} iRuthe = 4.1192 yards. --R"*- xMeae = { ^^^^Ja^^taS. Used by Miners, I Lachterlinie = .0927 inch. I o Lachterlinien i Lach terzoU = .9268 inch. 10 Lachterzoll i Achtel = .7723 foot 8 Achtels, or I t i.*. a jx 5 p^gg I I Lachter = 2.0596 yards. 9 Fuss i Spanne =6.1788 yards. Surveyor^ Measure, I Scrupel = .0148 inch. 10 Scrupel I Linie = .1483 inch. I o Linien i Zoll = 1.4828 inches. 10 ZoU I Land-Fuss = 1.2356 feet. 10 Land-Fuss i Ruthe =4.1192 yards. 2000 Ruthen i Meile = 4.6809 miles. II. Prussian Measures of Surface. I Square Linie = .00736 square inch. 144 Square Linien i Square Zoll - 1.0603 square inches. 144 Square Zoll i Square Fuss = 1.0603 square feet 144 Square Fuss i Square Ruthe =16.967 square yards. 180 Square Ruthen... i Morgen = .63103 acre. 30 Morgen i Hufe = 18.931 acres. III. Prussian Measures of Volume. Cubic Measure, I Cubic Linie = .000632 cubic inch. 1728 Cubic Linien.... i Cubic Zoll = 1.092 cubic inches. 1728 Cubic Zoll I Cubic Fuss = 1.092 cubic feet 1728 Cubic Fuss I Cubic Ruthe =69.893 cubic yards. For measuring stone and brickwork, earth, peat, fascines, and firewood, the following are u§e4 ;— PRUSSIA. — CAPACITY, WEIGHTS. 1 63 I Cubic Klafter, or ) . , . . . 108 Cubic Fuss ; - " 7.93 cubic feet 4ji Klafters i Haufe =530.70 I Schachnithe (in architecture) 144 Cubic Fuss = 157.25 99 IV. Prussian Measures of Capacity. Dry Measure, I Maasche = •7560 quart. JSS'"'"} ^""^^ = 3-4. quarts. 4 Metzen .. i Viertel = 3.0242 gallons. 4 Viertel, or ) ^ Scheffel = / ^'S'^' bushels, or 48 Quarts ) ( i. 941 cubic feet. 4 Scheffeln i Tonne = 6.0484 bushels. if 2S?r } ' Maker = 2..68rs quarters. 6jSeSn°'} '^' = I X.3407 quartets. The Tonne in the table is the measure for salt, lime, and charcoal. A Tonne of flax-seed is 2.354 Scheffeln. Liquid Measure (for Wine and Spirits). 32 Cubic ZoU I Ossel = 1.0079 pints. 2 Ossel I Quart = 1.0079 quarts. i:§Slf'" } ^^''^ = 7.559 gallons. 2 Ankers i Eimer = 15. 118 2 Elmers i Ohm = 30.237 ffS°' } ^Oxhoft =45.355 V. Prussian Measures of Weight. I Com = 4. 1 15 grains. 10 Corns I Cent = .09406 dram. 10 Cents I Quentche = .9406 dram. 10 Quentchen i Loth = ,588 ounce. 30 Loth I Zollpfund = 1. 1023 pounds. 1 00 Zollpfund I Centner = 110.23 pounds. 20 Zollpfund I Stein = 22.046 pounds. 330.69 pounds, or 2.506 hundredweights. 4409.2 pounds, or 3 Centners i Schiffspfund = ^ 40 Centners I x Schiffslast = { '^'^"^-g^g™; The Tonne of coals is 2270 pounds avoirdupois, or 1.013 tons. 1 64 WEIGHTS AND MEASURES. KINGDOM OF BAVARIA.— Old Weights and Measures.— Tables No. 46. * I. Bavarian Measures of Length. I Linie = .6798 inch. 12 Linien i ZoU = .95756 inch. 12 Zoll I Fuss = .95756 foot 6 Fuss i Klafter=5.74536 feet. 10 Fuss i Ruthe =9.5756 feet In surveying, the Fuss is divided into 10 Zoll, and i Zoll into 10 Linien. The EUe contains 2 Fuss lo^ Zoll, = 2.733 ^^^^ II. Bavarian Measures of Surface. I Square Zoll = .91692 square inch. 144 Square Zoll .... i Square Fuss = .91692 square foot 100 Square Fuss ... i Square Ruthe = 10.188 square yards. 400 Square Ruthen / ' Tagwerk Morgen, ) ^ ( 4075.188 square yards, or ^ I or Juchert J | .842 acre. III. Bavarian Measures of Volume. I Cubic Zoll = .878 cubic inch. 1728 Cubic Zoll I Cubic Fuss= .878 cubic foot 126 Cubic Fuss (6x6 xyA Fuss) i Klafter = / "^'•^28 cubic feet or ' ( 4.097 cubic yards. IV. Bavarian Measures of Capacity. Dry Measure, I Dreisiger= .12745 peck. 4Dreisigers i Maassl = .12745 bushel. 4 Maassls i Viertel = .5098 bushel. 2 Viertel i Metze =1.0196 bushels. 6 Metzen i Schaffel =6.1176 bushels. 4 SchafTel i Muth =3.0588 quarters. Liquid Measure, I Maaskanne= .23529 gallon. 64 Maaskannen i Eimer = 15.05856 gallons. 25 Eimer i Fass =376.464 gallons. The Schenk-Eimer, ordinarily used in die Wine trade, contains only 60 Maaskannen, equal to 14. 1 174 imperial gallons. V. Bavarian Measures of Weight. I Quentchen= .15433 ounce. 4 Quentchen i Loth = .6173 ounce. 32 Loth I Pfund = 1.23457 pounds. 100 Pfund I Centner = [ ^^3-457 pounds, or ( r.io2 hundredweights. WiJRTEMBEKG.— LENGTH, SURFACE, ETC. 165 KINGDOM OF WURTEMBERG.— Old Weights and Measures.— Tables No. 47. I. WiJRTEMBERG MEASURES OF LENGTH. I Punkte = .01128 inch. 10 Punkte I Linie = .1128 inch. 10 Linien i ZoU = 1. 128 inches. 10 Zoll I Fuss = .93995 foot 10 Fuss I Ruthe = 9.3995 feet. 2.144 Fuss I Elle = 2.015 feet 6 Fuss I Klafter = 5.6397 feet 26,000 Fuss I MeUe = i 8146.25 yards, or ' ( 4.6285 miles. II. WiJRTEMBERG MEASURES OF SURFACE. 1 Square Zoll = 1.272 square inches. 100 Square ZolL i Square Fuss = -8835 square foot 100 Square Fuss i Square Ruthe = 88.3506 square feet 384 Square Ruthen... i Moigen = { 3769-626 square yards, or III. WiJRTEMBERG MEASURES OF VOLUME. I Cubic Linie = .001434 cubic inch. 1000 Cubic Ltnien i Cubic Zoll = 1.434 cubic inches. 1 000 Cubic Zoll I Cubic Fuss = . 83045 cubic foot 144 Cubic Fuss I Cubic Klafter =119.583 cubic feet IV. WiJRTEMBERG MEASURES OF CAPACITY. I Dry Measure, I Viertlein = .305 pint 4 Viertlein i Ecklein =1.219 pints. 8 Ecklein i Vierling - 1.2 19 gallons. 4 Vierling i Simri =4.876 gallons. 8 Simri i Scheffel = 4.876 bushels. Liquid Measure, I Quart or Schoppen = .4043 quart. 4 Quarts i Helleich Maass = 1.6173 quarts. I o Helleich Maass i Irai = 4-0433 gallons. 16 Imi I Eimer = 64.6928 gallons. 6 Eimer i Fuder =388.1568 gallons. V. WiJRTEMBERG MEASURES OF WEIGHT. I Quentchen = .1289 ounce. 4 Quentchen i Loth . = .5156 ounce. 32 Loth I Light Pfund = 1.03115 pounds. \V, S?.7huS*.."} ^ C»^- = -^^396 pounds. 100 Light Pfund = 103. 115 pounds. 1 66 WEIGHTS AND MEASURES. KINGDOM OF SAXONY.— Old Weights and Measures.— Tables No. 48. I. Saxon Measures op Length. I Linie = .07742 inch. • 2 Linien i ZoU = .9291 inch. 12 Zoll I Fuss = .9291 foot 2 Fuss I EUe = 1.8582 feet 2 Ellen I Stab = 3.7165 feet 15 Fuss, 2 Zoll I Ruthe (Land Measure^ = 4.6972 yards. 16 Fuss I Ruthe (Road Measure) = 4.9553 yards. I Lachter (Mining) = 2.1873 yards. 1324.987 Ellen I Meile Post = 4.6604 miles. II. Saxon Measures of Surface. I Square Zoll = .8632 square. inch. 144 Square Zoll i Square Fuss = .8632 square foot 300 Square Ruthen i Acker = i . 4865 acres. III. Saxon Measures of Volume. I Cubic Zoll = .8021 cubic inch. 1728 Cubic Zoll I Cubic Fuss = .8021 cubic foot 108 Cubic Fuss I Klafter = 86.624 cubic feet 3 Klafter i Schragen =259.873 cubic feet The. Klafter is 6 Fuss by 6 Fuss by 3 Fuss. The Schragen is used in the measurement of firewood. IV. Saxon Measures of Capacity. Dry Measure, I Maasche = 1.4463 quarts. 4 Maaschen i Metze = 1.4463 gallons. 4 Metzen i Viertel = 5.7852 gallons. 4 Viertel i SchefTel = 2.8926 bushels. 12 SchefFel i Malter =34.7124 bushels. 2 Malter i Wispel = 69.4249 bushels. Liquid Measure, I Quartier = .2059 pint 4 Quartier i Nossel = .8237 pint 2 Nossel I Kanne = 1.6474 pints. 36 Kannen i Anker = 7.4237 gallons. 2 Anker i Eimer = 14.8262 gallons. 3 Eimer i Oxhoft = 44.4687 gallons. 6 Eimer i Fass or Barrel = 88.9374 gallons. V. Saxon Measures of Weight. The old Saxon measures of weight are the same as those of Prussia. BADEN. — LENGTH, SURFACE,- ETC 1 67 GRAND DUCHY OF BADEN.— Old Weights and Measures.— Tables No. 49. I. Baden Measures of Length. I Punkte = .0118 inch. 10 Punkte I Linie = .iiSi^jich. 10 Linien i ZoU = 1. 181 inches. 10 ZoU I Fuss = .9842 foot 2 Fuss I EUe = 1.9685 feet 10 Fuss I Ruthe = 9.8427 feet 6 Fuss I Klafter = 5.9055 feet 1 48 1 4.8 1 5 Fuss I Stunde =4860.59 yards. 2 Stunden i Meile = 5.5234 miles. II. Baden Measures of Surface. I Square ZoU = i-395i square inches. 100 Square ZoU i Square Fuss = .9688 square foot 100 Square Fuss i Square Ruthe = 10.7643 square yards. 100 Square Ruthen... i Viertel = 1076.43 square yards. 4 ^i-tel X Morgan = { ^^^l^^S:/"'^' " III. Baden Measures of Volume. I Cubic Fuss = .95335 cubic foot X 44 Cubic Fuss i Klafter = 137.28 cubic feet IV. Baden Measures of Capacity. Liquid Measure. I Glass - 1.0563 gills. 10 Glass I Maass = 1.3204 quarts. 10 Maass .*. i Stutze = 3.3014 gallons. 10 Stutzen I Ohm = 33.014 gallons. 10 Ohm I Fuder = 330.14 gallons. Dry Measure. I Becher = .2643 pint 10 Becher i Maasslein = .1652 peck. 10 Maasslein i Sester = .4127 bushel. 10 Sester i Maker = 4.1268 bushels. 10 Malter i Zuber = 41.2679 bushels. V. Baden Measures of Weight. I As = .7716 grain. 10 As I Pfennig = 7.716 grains. 10 Pfennig...., i Centas = .1764 ounce. 10 Centas i Zehnling = 1.7637 ounces. 10 Zehnling i Pfund = 1.1023 pounds. 100 Pfund iCentner = 110.230 pounds. 1 68 WEIGHTS AND MEASURES. THE HANSE TOWNS. — OlD WEIGHTS AND MEASURES. — Tables No. 50. HAMBURG. — ^Weights and Measures. I. Hamburg Measures of Length. I Acht'el = .1175 inch. 8 Achtel I ZoU = .9402 inch. 12 ZoU I Fuss = . .9402 foot. 2 Fuss i Elle = 1.8804 feet 6 Fuss i Klafter, or Faden= 5.6413 feet 14 Fuss i Marsch-Ruthe = 13.1629 feet 16 Fuss I Geest-Ruthe = 15.0434 feet The Hamburg Elle above is used for silk, linen, and cotton goods. The Brabant Elle is equal to i V5 Hamburg Elle; and 4 of them are reckoned equal to 3 yards. The Prussian Ruthe is also used. The Prussian Fuss is used in surveying. II. Hamburg Measures of Surface. I Square ZoU = .8840 square inch, 144 Square ZolL... i Square Fuss = .8840 square foot- 196 Square Fuss... i Square Marsch-Ruthe = 173.26 square feet 256 Square Fuss... i Square Geest-Ruthe = 226.30 square feet 200 Square Geest- ) o^^effel Oe^t T^nd - i 5028.98 square yards, or Ruthen / ^ ^^^^^^ oeest-l^na - ^ ^^^^ ^^^^^ 600 Sq. Marsch- ) ^ ^ ^ f ii55o-93 square yards, or Ruthen... J ^ ( 2.386 acres. III. Hamburg Measures of Volume. I Cubic Zoll = .8311 cubic inch. 1728 Cubic 2k)ll I Cubic Fuss = .8311 cubic foot 88.9 Cubic Fuss.... I (Cubic) Klafter = 73.88 cubic feet 1 2 o Cubic Fuss i Tehr =99-73 cubic feet IV. Hamburg Measures of Capacity. Liquid Measure. I Ossel = .09965 gallon. 2 Ossel I Quartier = .1993 gallon. 2 Quartier i Kanne = .3987 gallon. 2 Kannen i Stubchen =-• .7974 gallon. I Stubchen i Viertel = 1.5947 gallons. 4 Viertel i Eimer ^ 6. 3 7 88 gallons. 5 Viertel i Anker = 7.9735 gallons. 6 Eimer i Tonne = 38.2 728 gallons. 4 Anker i Ohm = 31.8940 gallons. 6 Anker i Oxhoft = 47.8410 gallons. 6 Ohm I Fuder, or Tonneau= 191.3640 gallons. The above are measures for Wines and Spirits. For Beer^ there are three sizes of Tonne, containing respectively 48, 40, and 32 Stubchen. HAMBURG. — WEIGHTS. 169 Dry Measure. I Small Maass = .0236 bushel 2 Small Maass i Large Maass = .0473 bushel. 4 Laige Maass i Spint = .1890 bushel. 4 Spint I Himten = .7560 bushel. 2 Himten i Fass = 1.5121 bushels. 2 Fass I Scheffel = 3.0242 bushels. 10 Scheffeln i Wispel • = 30.2416 bushels. 3 Wispel I Last =90.7248 bushels. For barley and oats, the Scheffel contains 3 Fass. V. Hamburg Measures of Weight. I Half Gramme = .0011 pound =.5 gramme. 10 Half Grammen i Quint = .01102 pound =5 grammes. 10 Quinten i (New) Unze = .11023 pound =50 „ 10 (New) Unzen.. i (New) Pfund = i.io232pounds = 5oo „ 100 (New) Pfund i Centner = 110.232 pounds = 50 kilog. 60 Centners. i (Commercial) Last = { ^or^^ q^.? ^^^ =3000 kilog. This, it is apparent, is a metric system of weights, which was comparatively recently introduced and adopted at Hamburg. It is now, of course, over- ruled by the French metric system enforced for the German Empire. BREMEN. — Old Weights and Measures. The Fuss is equal to 11.392 inches, and the Klafter is equal to 5.696 feet. The Morgen = .6368 acre. The principal measures for wines and spirits are the Viertel =1.56 gallons; the Anker = 5 Viertels = 7.80 gallons; the Oxhoft = 46.80 gallons. The Scheffel, for dry goods = 2.0388 bushels. The old weights are the same as those of Hamburg. LUBEC. — Old Weights and Measures. The Fuss is equal to 11.324 inches. The Viertel =1.60 gallons; the Anker = 8 gallons ; the Oxhoft = 48.04 gallons. The Scheffel, for dry goods, = .9545 bushel. The old Pfund =1.0725 pounds, and the Centner = 1.0725 cwts. GERMAN CUSTOMS UNION.— Old Weights and Measures.— Table No. 51. Centner 110.23 pounds (50 kilogrammes). Ship-Last of timber about 80 cubic feet. Scheffel 1.512 bushels. Klafter 6 feet. In Oldenbuig, Hanover, Brunswick, Saxe-Altenbourg, Birkenfeld, Anhalt, Waldeck, Reuss, and Schaumburg-I^ippe, the old system of weights is the same as that of Prussia. I/O WEIGHTS AND MEASURES. AUSTRIAN EMPIRE. — WEIGHTS AND MEASURES. — ^Tables No. 52. I. Austrian Measures of Length. I Punkte = .0072 inch. 12 Punkte I Linie = .0864 inch. 12 Linien i 2k)ll = 1.0371 inches. 12 Zoll I Fuss = 1.037 1 feet 2 Fuss i EUe = 2.0742 feet 6 Fuss I Klafter = 6.2226 feet 4000 Klafter i Meile(post) = { ^^^J. J^^^. II. Austrian Measures of Surface. I Square Zoll = 1,0756 squareinches. 144 Square Zoll i Square Fuss = 1.0756 square feet. 36 Square Fuss i Square Klafter = | 38.7"5 square feet, or ^ ^ -^uiu^ A ««Li.j ^1 4.3025 square yards. 8 }4 Square Klafter, or ) « -d ^i. - o- -j 300 Square ^uss } i Square Ruthe = 35.854 square yards. 64 Square Ruthen i Metze = 2294.7 square yards. 3 Metzen, or ) * , _ f 6884 square yards, or 1600 Square Klafter J -^ " ( 1.4223 acres. III. Austrian Measures of Volume. Cudic Miosure, I Cubic Zoll = I. "55 cubic inches, 1728 Cubic Zoll I Cubic Fuss = i-ii55 cubic feet 216 Cubic Fuss.... i Cubic Klafter = [ "4^-94 cubic feet or \ 8.924 cubic yards. IV. Austrian Measures of Capacity. Dry Measure. iProbmetzen = ( '""SpinljOr ( 3.665 cubic inches. 8 Probmetzen i Becher = .8460 pint 4 Becher i Futtermassel = 1.6920 quarts. 2 Futtermassel i Muhlmassel = [ ^'tf ^T ^^' ^^ \ .8460 gallon. 2 Muhlmassel i Achtel = 1.6920 gallons. 2 Achtel I Viertel = [ 3-384o gallons, or ( .4230 bushel. 4 Viertel i Metze = 1.69 18 bushels. 30 Meuen i Muth =/ So. 7536 bushels, or ^ \ 8.3442 quarters. AUSTRIAN EMPIRE. — CAPACITY, WEIGHTS. 171 Liquid Measure. I Pfiflf 2Pfiflf iSeidel = 2 Seidel i Kanne = 2 Kannen i Mass = 10 Mass I Viertel = 4 Viertel i Eimer = 32 Eimer.... i Fuder = 1.246 gills, or 10.781 cubic inches. 2.491 cubic inches, or .6229 pint 1.2457 pints. 1.2457 quarts. 3. 1 143 gallons. 12.4572 gallons. 398.6304 gallons. V. Austrian Measures of Weight. {270.1 grains, or .6173 dram. 2.4694 drams. 9.8776 drams, or .6173 ounce. 1.2347 ounces. 4.9388 ounces. 9.8776 ounces, or .6173 pound avoirdupois. = 1.2347 pounds avoirdupois. _ (123.47 pounds avoirdupois, or ~ ( 1. 1024 hundredweights. In 1853, a pfimd of 500 grammes, with decimal subdivisions, was adopted tor customs and fiscal purposes. I Pfenning 4 Pfenning i Quentchen 4 Quentchen... i Loth 2 Loth I Unze 4 Unzen i Vierdinge 2 Vierdinges... i Mark I.. I Pfund 100 Pfimd I Centner 2 Marks, or 16 Unzen RUSSIA. — ^Weights and Measures. — ^Tables No. 53. L Russian Measures of Length. English Equivalent. I Vershok = 1.75 inches. 16 Veishoks i Arschine = 28 „ 3 Arschines i Sajene = 7 feet. ( 3500 feet, or 500 Sajenes i Verst = < ii66^ yards, or ( 0.6629 mile. The Fuss, or Russian foot, is 13.75 inches; but, since 1831, the English foot of 1 2 inches has been used as the ordinary standard of length, each inch being divided into 12 parts. I Lithuanian Meile 5-5574 English miles. I Rhein Fuss, used in calculating ) t? r 1. r * export duties on timber / 1^2 WEIGHTS AND MEASURES. II. Russian Measures of Surface. o A I, .. f 784 square inches, or I Square Arschine = | ^ J ^ ^^^ ^^ 9 Square Arschines.. i Square Sajene = | ^^.^J^u^S'y^s. 2400 Square Sajenes i Desatine -\ ^' ; ^ o^^r^c ' III. Russian Measures of Capacity. Liquid Measure, I o Tscharkeys i Kruschka = 1.0820 quarts. 1 00 Tscharkeys i Vedro = 2. 7049 gallons. 3 Vedros i Anker = 8.1 147 „ tz v! AnkeS } •••• ' Sarokowaja Boshka = 108.196 Dry Measure (Grain). I Gamietz = 2.885 quarts. 2 Gamietz i Tschetwerka = i .4424 gallons. 4 Tschetwerkas... i Tschetwerik = .7213 bushel. 2 Tschetweriks.... i Pajak = 1.4426 bushels. 2 Pajaks I Osmin = 2.8852 „ 2 Osmins i Tschetwert* = 5.7704 „ ■« T-he*e«s . L». = { ".ijfi^JSKL * A Tschetv^ert is usually reckoned as 5ji^ bushels, and 100 Tschetwerts as 72 quarters^ though they are more exactly 72.1308 quarters. 100 quarters are equal to 138.637 Tschetwerts. For earthworks, masonry, &c., the Sajene is divided into tenths (dessiatka), hundredths (sotka), and thousandths (tisiatchka), which are used as a basis for lineal, superficial, and cubic measurements, similarly to the French metre with its sub-multiples. IV. Russian Measures of Weight. 96 Dolls I Zolotnick = < I Dolis = .68576 grain. 65.833 grains, or .1505 ounce. 3 2k)lotnicks... i Lotti = .4514 » 8 2k)lotnicks... i Lana = 1.2037 ounces. 12 Lanas, or \ i .90285 pound avoirdupois, or 32 Lottis, or > I Funt, or pound = < 14.446 ounces, or 96 2k)lotnicks j (6320 grains. 40 pounds I Pood = 36.114 pounds avoirdupois. 10 Poods I Berkovite = \ 36^-^4 pounds avoiidupois. or ( 3.224 hundredweights. 3 Berkovitz i Packen = 9.672 hundredweights. HOLLAND, BELGIUM, NORWAY, ETC. I73 62.0257 Poods I English ton. 2481.0268 Russian pounds i „ The Pood is commonly estimated at 36 pounds avoirdupois. The Nurembeig pound, used for apothecaries' weight, weighs 5527 grams, or about .96 pound troy. The Ship-Last is equal to 2 tons. The Caraty for weighing pearls and precious stones, is about 3 ^6 grains. HOLLAND. The metric system was adopted in Holland in 1819; the denominations corresponding to the French are as follows : — Ltngth, — Millimetre, Streep; centimetre, Duim; decimetre, Palm; metre, El; decametre, Roede; kilometre, Mijle. Surface, — Square millimetre, Vierkante Streep; square centimetre, Vier- kante Duim; and so on. Hectare, Vierkante Bunder. Cubic MecLsure, — Millistere, Kubicke Streep, and so on. Capacity, — Centihtre, Vingerhoed; decilitre, Maatje; liquid litre, Kan; dry litre, Kop; decalitre, Schepel; Hquid hectolitre. Vat or Ton; dry hectolitre. Mud or Zak; 30 hectolitres = i Last= 10.323 quarters. Weight, — Decigramme, Korrel; gramme, Wigteje; decagramme, Lood; hectogramme, Onze; kilogramme, Pond. BELGIUM. The Frenc3i metric S)rstem is used in Belgium. The name Livre is sabstitated for kilogramme, Litron for litre, and Aune for metre. NORWAY AND DENMARK. Weights and Measures. — Tables No. 54. I. Norwegian and Danish Measures of Length. I Linie = .0858 inch. 12 Linier i Tomme = 1.0297 inches. 12 Tommer i Fod = 1.0297 feet 2 Fod I Alen = 2.0594 „ 3 Alen, or 6 Fod 2 Favn, or 12 Fod 2,000 Roder, or ) j^^ ^ i 8237-77 yards, or > I Favn = 6.1783 „ I Rode = 12.3567 „ 24,000 Fod ( 4.68055 miles. 23,642 Fod I nautical mile= 4.61072 English miles. IL Norwegian and Danish Measures of Surface. 144 Square Linie i Square Tomme = 1.0603 square inches. 144 Square Tomme... i Square Fod = 1.0603 square feet 144 Square Fod i Square Rode = 16.966 square yards. 174 WEIGHTS AND MEASURES. III. Norwegian and Danish Measures of Volume. 1728 Cubic Linier i Cubic Tomme = 1.0918 cubic inches. 1728 Cubic Tomme.... i Cubic Fod = 1.0918 cubic feet The Favn of firewood measures 6x6x2 Fod= 72 cubic Fod = 78.60 cubic feet In forest measure it is 6)4 x 6j4 x 2 Fod = 84)^ cubic Fod = 92.26 cubic feet IV. Norwegian and Danish Measures of Capacity. Liquid Measure, I Paegle = .4248 pint 4 Paegle i Pot = 1.6991 pints. 2 Potter I Kande = 3.3983 „ 38 Potter I Anker = 8.0709 gallons. 136 Potter I Tonde = 28.885 * >> 6 Ankeme i Oxehoved = 48.4256 „ 4 Oxehoveder i Fad =193.7027 „ Dry Measure. I Pot = 1.699 1 pints. 18 Potter I Skeppe = 3.8232 gallons. 2 Skepper i Fjerdingkar = .9558 bushel. 4 Fjerdingkar i Tonde = 3.8231 bushels. 12 Tender i Laest =AS'^T^9 » V. Norwegian and Danish Measures of Weight. I Ort = 7.7163 grains. 10 Ort I Kvint = 77.163 „ 100 Kvinten i Pund = 1.1023 pounds. 100 Pund I Centner =110.23 „ 40 Centner i Last = i . 9684 tons. 52 Centner i Skip-Last = 2.5590 „ 16 Pund I Lispund = 17.63 7' pounds. 320 Pund I Skippund = 3.149 cwts. SWEDEN. — Weights and Measures. — Tables No. 55. I. Swedish Measures of Length. I Linie = .1169 inch. 10 Linier i Turn = 1. 1689 inches. 10 Tumer i Fot = 11.6892 „ 10 Fot I Stang = .9.7411 feet 10 Stanger i Ref = 32.4703 yards. 760 Ref I Meile - / ">6S9-3o8 yards, or ^^"^ ^^* ' ^"^^ - \ 6.6417 miles. 2 Fpt I Aln = 1-942 feet 6 Fot I Faden =^ 5.845 „ SWEDEN, SWITZERLAND. 1 75 II. Swedish Measures of Surface. loo Square Linier... i Square Turn = 1.3666 square inches. 100 Square Turner., i Square Fot = -9489 square foot 1 00 Square Fot i Square Stang = 3* 5 1 46 square yards. 100 Square Stanger i Square Ref = I •°54-^ square yards, or 4 Square Fot i Square Aln = 3-7956 square feet. 5.6 Square Ref:... r Tunnlar^d = { "^Lp^^S"'" " III. Swedish Measures of Volume. Cttbic Measure, I Cubic Turn =1.5972 cubic inches. 1000 Cubic Turner i Cubic Fot = .9263 cubic foot. 8 Cubic Fot I Cubic Aln = 7.4104 cubic feet Liquid and Dry Measure. 1 000 Cubic Linier i Cubic Turn = . 1 843 gill. 1 00 Cubic Turner i Kanna = 2. 3096 quarts. 10 Kanna i Cubic Fot = 5.774 gallons. 8 Cubic Fot I Cubic Aln =46.192 „ IV. Swedish Measures of Weight. I Kom = .6564 grain. 100 Kom I Ort = 2.4005 drams. 100 Ort I Skalpund = .9377 pound. -o Skalpur^d X Cer^tner = { 93-7739 pounds, or IOC Centner i Ny-Last = 4.1892 tons. A Pund, commercial, is .9377 pound. A Pund, freight, is .75016 pound. A Pund, miners* mark weight, is .8285 pound. A Pund, country town's mark weight, is .7891 pound. S^VITZERLAND. — ^WEIGHTS AND MEASURES. — Tables No. 56. I. Swiss Measures of Length. I Striche = .01181 inch. 10 Striche I Linie = .11811 „ 10 Linien i 2k)ll = 1.181 12 inches. 10 Zoll I Fuss = 11.81124 „ 2 Fuss i Elle = 1.9685 feet 6 Fuss i Klafter = 59056 ^ 10 Fuss i Ruthe = 98427 „ 1600 Ruthen .... i Schweizer-stunde, or Lien= < 2 o82^^iles 176 WEIGHTS AND MEASURES. II. Swiss Measures of Surface. I Square ZoU = 1-3947 square inches. 100 Square ZoU i Square Fuss = .9688 square foot 36 Square Fuss i Square Klafter = 34.8768 square feet 100 Square Fuss i Square Ruthe = 10.7643 square yards 400 Square Ruthen. . i Juchart = .8694 acre. 6400 Jucharten i Square Stunde = 5693.52 acres. 350 Square Ruthen i Juchart, of meadow land. 450 Square Ruthen i Juchart, of woodland. III. Swiss Measures of Volume. I Cubic Zoll = 1.6476 cubic inches. 1000 Cubic ZolL I Cubic Fuss = .9535 cubic foot 216 Cubic Fuss I Cubic Klafter = 7.6172 cubic yards. 1000 Cubic Fuss I Cubic Ruthe =35.3166 „ IV. Swiss Measures of Capacity. Dry Measure, I Imi = 1.3206 quarts. 10 Imi I Maass = .4127 bushel. 10 Maass i Malter = 4.1268 bushels. Liquid Measure, 2 Halbschoppen i Schoppen = 2.6412 gills. 2 Schoppen i Halbmaass = 1.3206 pints. 2 Halbmaass i Maass = 2.6412 „ 100 Maass i Saum = 33.015 gallons. V. Swiss Measures of Weight. I Quntii = 2.2048 drams. 4 Quntli I Loth = .5511 ounce. 2 Loth I Unze = 1. 1023 ounces. 16 Unzen i Pfund = 1. 1023 pounds. 100 Pfund I Centner = 110.233 pounds, or .9842 cwt. The Pfund is divided into halves, quarters, and eighths. It is also divided into ^00 Grammes, and decimally into Decigrammes, Centi* grammes, and Milligrammes. SPAIN. — ^Weights and Measures. — Tables No. 57. The French metric system was established in Spain in 1859. The metre is named the Metro; the litre, Litro; the gramme, Grammo; the are, Area; the tonne, Tonelada. The metric system is established likewise in the Spanish colonies. The old weights and measures are still largely used. SPAIN — LENGTH, SURFACE, ETC. 1 77 L Old Spanish Measures of Length. I Punto = .00644 inch. 12 Puntos I Linea = .07725 inch. 12 Lineas i Pulgada = .927 inch. 6 Pulgadas i Sesma = 5.564 inches. 2 Sesmas i Pies de Burgos = .92 73 foot. 3 Pies de Burgos i Vara = 2.782 feet 2 Varas i Estado = 5.564 feet 4 Varas i Estadal =11.128 feet 5000 Varas i Legua (Castilian) = 2.6345 miles. 8000 Varas i Legua (Spanish) = 4.2 151 miles. II. Old Spanish Measures of Surface. I Square Pies = .860 square foot 9 Square Pies i Square Vara = .860 square yard. 16 Square Varas i Square Estadal = 13.759 square yards. 50 Square Varas .... i Estajo =42.997 square yards. 576 Square Estadals. i Fanegada = 1.6374 acres. 50 Fanegadas. i Yugada = 81.870 acres. III. Old Spanish Measures of Capacity. Liquid Measure. I Capo = .888 gill. 4 Capos I Cuartillo = .111 gallon. 4 Cuartillos i Azumbre = .444 gallon. 2 Azumbres i Cuartilla = .888 gallon. ^ ^.« ( I ArrobaMayor,orCantara) ^ ^^ „«n^«« 4 CuartUlas... | (for wine) j " 3-552 gallons. 16 Cantaras i Mayo =56.832 gallons. The old measure for oil is the Arroba Menor= 2.7652 gallons. JDry Measure, I Ochavillo = .00785 peck. 4 Ochavillos i Racion = .0314 peck. 4 Raciones i Quartillo = . 03 1 4 bushel. 2 Quartillos i Medio = .0628 bushel. 2 Medios i Almude = .1256 bushel; 12 Amuerzas i Fanega = 1.5077 bushels. 1 2 Fanegas i Cahiz = 1 8.0920 bushels. IV. Old Spanish Weights. I Grano = .771 grain. 12 Granos i Tomin = 9.247 grains. 3 Tomines .... I Adarme = 27.74 grains. 2 Adarmes .... i Ochavo, or Drachma = .1268 ounce. 8 Ochavos i Onza = i.o 144 ounces. 8 Onzas i Marco = 8.1 154 ounces. 2 Marcos i Libra (Castilia a) = i . o 1 44 pounds. IOC Libras i Quintal = 101.442 pounds. 10 Quintals i Tonelada =1014.42 pounds. 19 178 WEIGHTS AND MEASURES. PORTUGAL. The French metric system of weights and measures v^as adopted in its entirety during the years 1860-63, and was made compulsory from the ist October, 1868, The chief old measures still in use are, the Libra = 1.012 pounds; Almude, of Lisbon = 3.7 gallons; Almude, of Oporto = 5.6 gallons ; Alquiere = 3.6 bushels; Moio = 2.78 quarters. ITALY. The French metric system is used in Italy. The metre is named the Metra; the are, Ara; the stfere, Stero; the litre, Litro; the gramme, Gramma; the tonneau m^trique, Tonnelata de Mare. The various old weights and measures of the different Italian States are still occasionally used. TURKEY. Length. — i Pike or Dri= 27 inches, divided into 24 Kerats; i Forsang = 3.116 miles, divided into 3 Berri; the Surveyor's Pik, or the Halebi = 27.9 inches; and 5J^ Halebis= i reed. Surface, — The squares of the Kerat, the Pike, and the Reed. The Feddan is an area equal to as much as a yoke of oxen can plough in a day. Capacity^ Dry, — ^The Rottol-i.411 quarts, contains 900 Dirhems; 22 Rottols= I Killow= 7.762 gallons, or .97 bushel, the chief measure for grain. Liquid. — i Oka= 1.152 pints; 8 Oke= i Almud= 1.152 gallons; i Rottol = 2.5134 pints; 100 Rottols= i Cantar = 31.417 gallons. Weights. — ^The Oke= 2.8342 pounds, divided into 4 Okiejehs, or 400 Dirhems of 1.81 drams; i Rottolo= 1.247 pounds; 100 Rottolos= i Cantar = 124.704 pounds. GREECE AND IONIAN ISLANDS. The French metric system is employed in Greece. The metre is named the Pecheus; kilometre, Stadion; are, Stremma; litre, Litra; gramme. Drachm^. 1% kilogrammes = i Mni; lyi Quintals = i Tolanton i}i Tonneaux= I Tonos= 29.526 cwts. In the Ionian Islands, whilst they were under the protection of Great Britain (1830 to 1864), the British weights and measures were those in use, with Italian names. The foot was named the Piede; the yard, the Jarda; the pole, the Camaco; the furlong, the Stadio; the mile, the Miglio. The gallon was the Gallone; the bushel, the Chilo; the pint, the Dicotile; the pound avoirdupois, the Libra Grossa; the pound troy, the Libra Sottile. The Talanto consisted of 100 pounds, and the Miglio of 1000 pounds. MALTA. In round numbers, 3^ Palmi= i yard; i Canna = 2 "/^ yards. The Salma = 4.964 acres. Approximately, 543 Square Palmi = 400 square feet; 16 Salmi = 71 acres. EGYPT. — LENGTH, SURFACE, ETC. 1 79 I Cubic Tratto = 8 cubic feet; 144 Cubic Palmi = 96 cubic feet; i Cubic Canna = 543 cubic feet Approximate weights: — 15 Oncie=i4 ounces; i Rotolo=i^ pounds; 4 Rotoli = 7 pounds; 64 Rotoli = i cwt. ; i Cantaro =175 pounds; i Quintal = 199 pounds; 64 Cantari = 5 tons. EGYPT. — ^Weights and Measures. — ^Tables No. 58. I. Egyptian Measures of Length. Pik, or cubit of the Nilometre 20.65 inches. Pik, indigenous 22.37 „ Pik, of merchandise 25.51 „ Pik, of construction 29.53 „ 6 Palms I Pik. 24 Kirats i Pik or Dr^, 4.73 Piks of construction... i Kassaba in surveying, =11.65 feet. 11. Egyptian Measures of Surface. I Square Pik = 6.055 square feet 22.41 Square Piks i Square Kassaba = 15.07 square yards 333.33 Square Kassaba, i Feddan = .9342 acre. III. Egyptian Measures of Capacity. * I Kadah = 1.684 pints. 2 Kadahs i Milwah = 6.735 » 2 MDwahs I Roobah = 1.684 gallons. 2 Roobahs '. i Kelah = 3.367 „ 2 Kelehs i Webek = 6.734 „ 6 Webeks i Ardeb - / ^0.404 gallons, or ^ ^^«^^*^ ^ ^^^^ - I 6.48 cubic feet The Guirbah of water (a government measure) is V15 cubic metre = 66*4 litres, or 11.772 cubic feet IV. Egyptian Measures of Weight. I Kamhah - .746 grain. 4 Kamhahs. i Kerat 16 Kerats i Dirhem = 1.792 drachms. 24 Kerats i Mitkal. SMitkals i Okieh. x^: Salt's"} ^R°"°l = -^S.! pound. 100 Rottols I Kantar =98.207 pounds. 400 Dirhems i Oke = 2.728 „ 36 Okes I Kantar =98.207 „ l80 WEIGHTS AND MEASURES. MOROCCO. Length, — ^The Tomin = 2.81025 inches; the Dra'a = 8 Tomins = 22.482 inches. Capacity, — ^The Muhd = 3.08135 gallons; the Sai = 4 Muhds= 12.3254 gallons. Weights, — The Uckia = 392 grains; the Rotal or Artal = 2o Uckieh = 1,12 pounds; the Kintar= 100 Rotales= 112 pounds. Oil is sold by the Kula = 3.3356 gallons. Other liquids are sold by weight TUNIS. Length, — ^The Dhrai, or Pike, is the unit of length. The Arabian Dhraa, for cotton goods =19.224 inches; the Turkish Dhrai, for lace = 25.0776 inches; the Dhrai Endaseh, for woollen goods = 26.4888 inches. The Mil Sah'ari = .9i49 mile. Capacity, — For dry goods the Sai= 1.2743 pint; 12 Saa=i Hueba = 6.8228 gallons. For liquids, the Pichoune = .4654 pint; 4 Pichounes=i Pot =1.8616 pints; 15 Pots = I Escandeau, and 4 Escandeaux=i Mill^role= 13.9623 gallons. ARABIA. The weights" and measures of Egypt are used in Arabia. CAPE OF GOOD HOPE. The standard weights and measures are British, with the excepfton of the land measure. To some extent, the old British and the Dutch measures are in use. The general measure of surface is the old Amsterdam Morgen^ reckoned equal to 2 acres; though the exact value is equal to 2.1 1654 acres. 1000 Cape feet are equal to 1033 British feet INDIAN EMPIRE. — ^WEIGHTS AND MEASURES. An Act " to provide for the ultimate adoption of an uniform system of weights and measures of capacity throughout British India " was passed in October, 187 1. The ser is adopted under the Act as the primary standard or unit of weight, and is a weight of metal in the possession of the Govern- ment, equal, when weighed in a vacuum, to one kilogramme. The unit of capacity is the volume of one ser of water at its maximum density, equiva- lent to the litre. Other weights and measures are to be multiples or sub- multiples of the ser, and of the volume of one ser of water. The following are the weights and measures in common use in India: — BENGAL — LENGTH, SURFACE, ETC l8l BENGAL. — ^Weights and Measures. — Tables No. 59. I. Bengal Measures of Length. I Jow, or Jaub = ]^ inch. 3 Jow I Ungulee = ^ „ 4 Ungulees i Moot = 3 inches. 3 Moots I Big'hath, or Span = 9 „ 2 Big'haths i Hit'h, or Cubit... = 18 „ 2 Hat'h I Guz = I yard. 2 Guz I Danda, orFathom = 2 yards. ^o- ^^^ ^ Coss = { --15t'i. 4 Coss I Yojan = 4.5454 miles. II. Bengal Measures of Surface. I Square Hat'h = 2.25 square feet. 4 Square Hit'hs i Cowrie = i square yard. 4 Cowries i Gunda = 4 square yards. 20 Gundas i Cottah = 80 „ 20 Cottahs I Beegah = i '^°° ^""f^ V^^' °' \ -3306 acre. For land measure, the following table is used for Government surveys : — I Guz = 33 lineal inches. 3 Guz I Baus,orRod= 8^ lineal feet. 9 Square Guz i Square Rod = 68 Vx6 square feet. 400 Square Rods..... i Beegah = { 3°*S Weyards, or III. Bengal Measures of Capacity. The Seer is a measure common to liquids and dry goods. It is taken at 68 cubic inches, or 1.962 pints, in volume. But it varies in different localities. 5 Seer= i Palli, and 8 Palli= i Maund, or 9.81 gallons. The Sooli = 3.065 bushels, and 16 Soolis = i Khahoon, or 49.05 bushels. IV. Bengal Measures of Weight. The Tola, or weight of a Rupee, 180 grains, is the unit of weight. I Tola =180 grains. 5 Tolas I ChittHk =900 „ 16 Chittiks I Seer = 2.057 pounds. 5 Seers i Passeeree = 10.286 „ 8 Passeerees i Maund = 82.286 „ MADRAS. — Weights and Measures. — Tables No. 60. I. Madras Measures of Length. The English foot and yard are used. The Guz is 33 inches. The Baum or fathom is about 6j4 feet. A Nilli-Valli is a little under ij^ miles. 7 Naili-Valli = I Kadam, or about 10 miles. The following are native measures : — 1 82 WEIGHTS AND MEASURES. 8 Torah i Vurruh = .4166 inch. 24 Vurmh i Mulakoli = 10 inphes. 4 Mulakoli i Dumna = 40 „ II. Madras Measures of Surface. The English acre is generally known. The native me^tsures are uncer- tain. In Madras and some other districts^ the following native measures are used : — I Coolie = 64 square yards. 4 Ve Coolies i Ground = 266^ square yards. 24 Grounds, or ) rown*** - / 6400 square yards, or 100 Coobes J (1.3223 acres. 16 Annas (each 400 yards), i Cawnie. III. Madras Measures of Capacity. I OUuck = .361 pint 8 OUucks I Puddee = 1.442 quarts. 8 Puddees i Mercil = 2.885 gallons. 5 Mercils i Parah =14.426 „ 80 Parahs i Garce = 18.033 quarters. This, though the legal system, is not used. The "customary" Puddee is still in general use; it has, when slightly heaped, a capacity of 1.504 quarts. The Mercil has a capacity of 3.0006 gallons; but, when heaped, it is equal to 8 heaped Puddees. The Seer-measure is the most common; its cubic contents are from 66 }4 to 67 cubic inches. IV. Madras Measures of Weight. I Tola = 180 grains. 3 Tolas I PoUum = 1.234 ounces. 8 PoUums I Seer = 9-874 „ 5 Seers i Viss = 3.086 pounds. 8 Viss I Maund = 24.686 „ 20 Maunds . Candy = { ^^J J^^ J^ '^' " In commerce, the Viss is reckoned as 3^ pounds; the Maund, 25 pounds; and the Candy, 500 pounds. BOMBAY. — ^\Veights and Measures. — Tables No. 61. I. Bombay Measures of Length. I Ungulee = 9/,6 inch. 2 Ungulee i Tussoo = i^ inches. . 8 Tussoos I Vent'h =9 „ 16 Tussoos I Hat'h =18 „ 24 Tussoos I Guz =27 „ The Builder's Tussoo = 2.3625 inches in Bombay; and i inch in Surat BOMBAY, CEYLON, BURMAH. 1 83 II. Bombay Measures of Surface. 34 Ve Square Hat'h... i Kutty = 9-3i75 square yards. 20 Kutties I Fund = 196.35 „ -P-nd ^ Beegah= {39^7^f-yf «'- 1 20 Beegah i Chahur = 97-368 acres. In the Revenue Field Survey, the English acre is used III. Bombay Measures of Capacity. I Tippree= .2800 pint. 2 Tipprees i Seer = .5600 „ 4 Seers i Pylee = 2.2401 pints. 16 Pylees i Parah = 4.4802 gallons. 8 Parahs i Candy = 35.8415 „ 25 Pamhs I Mooda = { '""^^^S g^Uons or •^ ( I-750I quarters. Another liquid measure is the Seer of 60 Tolas = 1.234 pints. In timber measurement in the Bombay dockyards, a Covit or Candi = 12.704 cubic feet. CEYLON. The British weights and measures are used. BURMAH. The English yard, foot, and inch are being adopted; also the English Measures of Capacity. Weights. — The Piakthah or Viss is 3.6 pounds, and contains 100 Kyats of 252 grains each. CHINA. — ^Weights and Measures. — Tables No. 62. I. Chinese Measures of Length. I Fun = .141 inch. 10 Fun I Tsun = i. 41 inches. 10 Tsun I Chih =14.1 „ 10 Chih I Chdng= 11.75 ^"^^t. 10 Ching I Yin =39.17 yards. The Chih of 1 4.1 inches is the legal measure at all the ports of trade. At Canton, the values of the Chih are as follows : — Tailor's Chih 14.685 inches. Mercer's Chih (wholesale) 1 4. 66 to 1 4. 7 2 4 inches. Mercer's Chih (retail) ^4*37 to 14.56 „ Architect's Chih 12.7 inches. At Pekin there are thirteen different Chihs. 1 84 WEIGHTS AND MEASURES. Distance, 5 Fun I Li = .486 inch. 10 Li )^Chih = .405 foot. 5 Chih I Pii = 4.05 feet. 360 Pii I Li =486 yards. 250 Lf I Tii (or Degree) = 69 miles. II. Chinese Measures of Surface. 25 Square Chih i Pii or Kung= 3.32 square yards 60 Kung I Kish =199.47 4 Kish I Mau =797-89 100 Mau I King = 16,485 acres. The chief land measure is the Mau, than which smaller areas are expressed decimally. III. Chinese Measures of Capacity. {Dry Measure,) I Koh = .0113 gallon. 5 Koh >^Shing = .0565 „ 10 Koh I Shing = .113 „ 10 Shing I Tau =1.13 gallons. Liquids are measured by vessels containing definite weights, as i, 2, 4^ and 8 Taels; also large earthen vessels holding 15, 30, and 60 Catties. See Table of Weights. IV. Chinese Measures of Weight. I L^ang or Tael = i '/a ounces. 16 L^ang I Kin or Catty = i V3 pounds. 100 fcin I Tan or Pecul = [ ^33-33 Pounds, or ( 1. 19 cwts. COCHIN-CHINA. Length, — The Thuoc, or cubit, 19.2 inches, is the chief unit of measure of length. It varies considerably for different places. The Li or mile is 486 yards; 2 Li make i Dam; and 5 Dam make i league =2.761 miles. Surface, — 9 Square Ngu make i Square Sao = 64 square yards. 100 Square Sao make i Square Mao = 6400 square yards, or 1.32 acres. Weights, — The smallest weight is the Ai = .0000006 grain. The weights ascend by a decimal scale, until 10,000,000,000 Ai are accumulated = I Nen = .8594 pound. The greatest weight is the Quan = 6875^ pounds. Capacity for Grain, — i Hao = 6»/g gallons. 2 Hao = i Shita=i2 4/j gallons. PERSIA. Length, — The Gereh = 2^ inches; 16 Gerehs= i Zer=38 inches. The Kadam or Step = about 2 feet; 12,000 Kadam = i Fersakh = about 4)^ miles. PERSIA, JAPAN. 185 Surfcue and Cubic Measures, — ^These are the squares and cubes of the lengths. Capacity (Dry Goods). — The Sextario = .o7236 gallon. 4 Sextarios = I Chenica; 2 Chenicas= i Capicha; 3^ Capichas= i CoUothun; 8 Collo- thun = I Artata= 1.809 bushels. liquids are sold by weight Weights, — ^The Miscal = 7i grains; 16 Miscals=i Sihr; 100 Miscals = I Ratal = 1.014 pounds; 40 Sihrs = i Batman (Maund) = 6.49 pounds; 100 Batman (of Tabreez)= i Karwar = 649.i42 pounds. JAPAN. — ^Weights and Measures. — Tables No. 63. I. Japanese Measures of Length. I Rin = .012 inch. 10 Rin I Boo = .120 inch. 10 Boo I Sun = 1.20 inches. 10 Sun I Shiaku = i foot. 3 Shiaku ^ Ken = i yard. 6 Shiaku i Ken - 2 yards. 60 Ken I Chu = 120 yards. _ J 4320 yards, or 2.454 miles. 36 Chu., I Ri = I Rough timber is sold by the Yama-Ken-Zau = d^ Sun. Cloth is measured by the Shiaku of 15 inches, with decimal sub-multiples. II. Japanese Measures of Surface. I Po = 4 square yards. 30 Po I Is'she = 120 square yards. I o Is'she I It'tau = 1 200 square yards. 10 Iftau. I Ifchoe = ( "°°° T"^ y^''*'' °'" I 2.4793 acres. The square Ken is the unit of square measure, equal to 4 square yards, III. Japanese Measures of Capacity. * Dzoku = .0000328 pint. 10 Dzoku I Ke = .000328 pint. 10 Ke I Sat = .000328 pint. 10 Sats I Sai = .00328 pint. 10 Sal I Shiaku = .03283 pint. 10 Shiaku i Goo = .3283 pint 10 Goo I Shoo = .4104 gallon. 10 Shoo I To = 4.104 gallons. xo To I Koku =41.04 gallons. 1 86 WEIGHTS AND MEASURES. IV. Japanese Measures of Weight. I Mo = .027 grain. 10 Mo I Rin = .2701 grain. 10 Rin I Fun = 2.701 grains. 10 Fun I Noihme = 27.006 grains. 4 Nomme i Riu = 108.026 grains. 40 Riu I Kiu = .6173 pound. JAVA. Length. — The Duim=i.3 inches. 12 Duims=i foot. The Ell = 27.08 inches. Surface, — The Djong of 4 Bahu = 7.015 acres. Capacity, for rice and grain. — The measures are in fact measures of definite weights, i sack = 61.034 pounds; 2 sacks =1 Pecul; 5 Peculs = I Timbang = 5.45 cwts.; 6 Timbang= i Coyaii = 32.7 cwts. For Hquids: The Kan = .328 gallon; 388 Kans=: 1 Leager= 127.34 gallons. Weights. — The Tael = 593.6 grains; i6Taels=^i Catty = 1.356 pounds: 100 Catties = i Pecul- 135.63 pounds. UNITED STATES OF AMERICA. Length, — The measures are the same as those of Great Britain. In Land Surveying, the unit of measurement is the chain, and it is deci- mally subdivided. In City Measurements, the unit is the foot, and it is decimally subdivided. In Mechanical Measurements, the unit is the inch, and it is divided into a hundred parts. Surface, — The measures are the same as those of Great Britain. Capacity, — The measures of capacity for dry goods and for liquids are the same as the old English measures. 'Fhe standard U. S. gallon is equal to the old English wine gallon, or 231 cubic inches; it contains 8^ pounds of pure water at 62° F. Dry Measure, — Table No. 64. I gill. = .96945 imperial gill. 4 gills I pint = .96945 imperial pint 2 pints I quart ~ 1.9388 „ pints. 4 quarts i gallon = .96945 „ gallon. 2 gallons I peck = 1.9388 „ gallons. 4 pecks I bushel = -96945 „ bushel. 4 bushels I coomb =3-8777 „ bushels. 2 coombs I quarter = .96945 „ quarter. 5 quarters i wey or load =4.8472 „ quarters. 2 weys I last =9-6945 „ quarters. For the Wine and Spirit Measures, and the Ale and Beer Measures, see the Old Measures of Great Britain, page 139. I cord of wood =128 cubic feet = (4 feet x 4 feet x 8 feet). Weights, — The Weights are the same as those of Great Britain. (See page 140.) BRITISH NORTH AMERICA, ETC. 1 8/ There are, in addition, the Quintal or Centner of loo pounds; and the New York ton of 2000 pounds, which is also used in most of the States, The old hundredweight and ton are for the most part superseded by the quintal and the New York ton. The French metric system of weights and measures has been legalized concuirendy with the old system. BRITISH NORTH AMERICA.— WEIGHTS AND MEASURES. Until the 23d May, 1873, the standard measures of length and surface, and the weights, were the same as those of Great Britain; whilst the measures of capacity were the old British measures for dry goods, for wine, and for ale and beer. At the above-named date a new and uniform system of weights and measures came into force, in which the imperial yard, pound avoirdupois, gallon, and bushel, became the standard units, and the imperial system was adopted in its integrity, with two important exceptions : that the hundredweight of 112 pounds, and the ton of 2240 pounds were abolished; and the hundredweight was declared to be 100 pounds, and the ton 2000 pounds avoirdupois, — thus assimilating the weights of Canada to those of the United States. The French metric system of weights and measures has been made permissive concurrently with the standard weights and measures. MEXICO. The weights and measures are the old weights and measures of Spain. CENTRAL AMERICA AND WEST INDIES. WEST INDIES (British). The weights and measures are the same as those of Great Britain. CUBA. The old weights and measures of Spain are in general use. For engineer- ing and carpentry work the Spanish, English, and French measures are in use. The French metric system of weights and measures is legalized, and is used in the customs departments. GUATEMALA AND HONDURAS. The weights and measures are the old weights and measures of Spain. BRITISH HONDURAS. In British Honduras, the British weights and measures are in use. COSTA RICA. The old weights and measures of Spain are in general use. But the introduction of the French metric system is contemplated. l88 WEIGHTS AND MEASURES. ST. DOMINGO. The old Spanish weights and measures are in general use. The French metric system is coming into use. SOUTH AMERICA. COLOMBIA. The French metric system was introduced into the Republic in 1857, and is the only system of weights and measures recognized by the govern- ment. In ordinary commerce, the Oncha, of 25 lbs., the Quintal, of 100 lbs., and the Carga, of 250 lbs., are generally used. The libra is 1. 102 pounds. The yard is the usual measure of length. VENEZUELA. The system and practice are the same as those of Colombia. ECUADOR. The French metric system became the legal standard of weights and measures on the ist January, 1858. GUIANA. In British Guiana, the weights and measures are those of Great Britain. In French Guiana or Cayenne, the ancient French system is practised In Dutch Guiana, the weights and measures of Holland are employed. BRAZIL. The French metric system, which became compulsory in 1872, was adopted in 1862, and has since been used in all official departments. But the ancient weight§ and measures are still partly employed. They are, with some variations, those of the old system of Portugal. Length, — ^The Line = .09 11 inch, and is divided into tenths. The PoUe- gada = 1.0936 inches. The Pd = 1 3. 1 236 inches, or ^j^ metre. The Vara = 1.2 1 5 yards; and ij^ Varas = the geometrical pace =1.8227 yards. The Milha= 1.2965 miles; and 3 Milhas = i Legoa = 3.8896 miles. 6 yards are reckoned equal to 5 Varas. Surface, 64 Square Pollegadas... i Square Palmo = .5315 square foot 25 Square Palmos i Square Vara =1.4766 square yards. 4 Square Varas i Square Braga = 5. 9063 „ 4840 Square Varas i Geira = 1.4766 acres. Capacity {Dry Goods), — The Salamine = .38o8 gallon; 2 Salamines = \ Oitavo; 2 Oitavo = i Quarto; 4 Quartas=i Alqueiro = .38o8 bushel; 4 Alqueiras = i Fangas; 15 Fangas = i Moio = 2.8560 quarters. Liquids, — The Quartilho = .6i4i pint; 4 Quartilhos= i Canada; 6 Cana- das = I Pota or Cantaro; 2 Potas = i Almuda = 3.6846 gallons. PERU, CHILI, BOLIVIA, ETC. 1 89 Weights. — The Arratel= 1.0119 pounds, is divided into 16 Ongas, and then into 8 Oitavos. 32 Arratels=i Arroba; 4 Arrobas = i Quintal = 129.5181 pounds; and 13 J^ Quintals = 1 Tonelada= 15.6116 cwts. There is also the Quintal of 100 Arratels; Ships' freight is reckoned by the English ton = 70 Arrobas. PERU. The French metric system was established in i860, but is not yet gener- ally used. The weights and measures in common use are : — ^The ounce = 1.014 ounce; the Libra=i.oi4 pound; the Quintal = 101.44 pounds; ^^ Arroba =25.36 pounds, or 6.70 gallons; the gallon = .74 imperial gallon; the Vara = .92 7 yard; the square Vara = .85 9 square yard. CHILL The French metric system has been legally established; but the old weights and measures are still in general use. These are the same as those of Peru. BOLIVIA. The weights and measures are the same as the old weights and measures of Peru and Chili. ARGENTINE CONFEDERATION. The French metric system has recently been established. The old weights and measures are commonly used: — th^ Castilian standards of the old Spanish system. The Quintal = loi. 4 pounds; the Arroba = 25.35 pounds; the Fanega=i.5 bushels. URUGUAY. The French metric sjrstem was established in 1864. The old weights and measures are the same as those of the Argentine Confederation. The weights and measures of Brazil are in general use. PARAGUAY. The weights and measures are the same as the old ones of the Argentine Confederation. AUSTRALASIA. In New South Wales, Queensland, Victoria, South Australia, West Australia, Tasmania, and New Zealand, the legal weights and measures are the same as those of Great Britain. But the old British measures of capacity are also much used. In land measurement, a "section" is an area equal to 80 acres. MONEY. GREAT BRITAIN AND IRELAND. Coins. Material. Weight. Grains. j{d, farthing bronze. }id. halfpenny do. 4 farthings i penny do. 3//. threepenny piece silver. 4i/. groat, or fourpenny piece do. 6d. sixpence do. 12 pence i shilling do. 2 shillings i florin do. 2j^x. I half-crown do. lox. I half-sovereign gold.. 20s, I sovereign, or pound sterling do. • 43- 750 . 87.500 .145-833 . 21.818 . 29.091 • 43.^36 . 87.273 .174.545 .218.182 . 61.6372 .123.2745 The bronze coins are made of an alloy of copper, tin, and zinc; the silver coins contain 92 J^ per cent, of fine silver, and Tj4 per cent of alloy; the gold coins, 91^ per cent, of fine gold, and 8^ per cent, of alloy. The Mint price of standard gold is ;£'3, ijs, ioJ4//. per ounce. One pound weight of silver is coined into 66 shillings. The intrinsic value of 22 shillings is equal to £1 sterling. The intrinsic vdue of 480 pence is equd to £1 sterling. FRANCE.— Money. Vxoo firanc . V50 franc Vao franc Copper, Coins. Weight. Value in English Money. Grammes. £ S, a, 1 centime i o 2 centimes 2 o 5 centimes (j^«) 5 o '/xo franc 10 centimes (gr^jj^w)... 10 .0 o o o o xo 'A I 2 5 franc, franc, franc. Silver, 20 centimes i... 50 centimes 2.5. 100 centimes 5 .0 .0 .0 o 2 o aM, o g}4 francs 10 francs 25 more exactly 9.524//. .0 .0 I 3 7 GERMANY, HANSE TOWNS. I9I GoU. Grammes. JZi ^' "• 5 francs. 1*61290 o 3 11^ 10 francs. 3*22580 o 7 ii^ 20 francs (Nj^)oleon)... 6*45161 (99*56 grains)...o 15 10^ 50 francs 16*12902 i 19 8^5 100 francs 32*25805 3 19 4 4/,^ The English value is calculated at the rate of 25 francs 20 centimes to j£i. The bronze coins consist of an alloy of 95 parts of copper, 4 of tin, I of zinc. The standard fineness of the gold pieces, and of the silver 5-franc pieces is 90 per cent., with 10 per cent of copper; of the other alver coins, 83.5 per cent.; and of the bronze coins, 95 per cent. GERMANY.— Money. The following system of currency was established throughout the German Empire in 1872: — English Value. s, d. I Pfennig = o .1175 ID Pfennig i Groschen = o 1.175 10 Groschen i Mark = o iij^ 10 Marks (gold).. = 9 9^ 20 Marks (gold)... = 19 7 The 20-mark gold piece weighs 122.92 grains, and the standard fineness of the gold pieces is 90 per cent of gold. Before 1872, accounts were reckoned in the following currency in North Germany : — s, d, 12 Pfennig i Silbergroschen = i i '/s 30 Silbergroschen i Thaler = 30 In South Germany: — 4 Pfennig i Kreutzer = o ^ 60 Kreutzers. i Florin = 18 HANSE TOWNS.— Money. The monetary system is that of the German Empire. Hamburg. — ^According to the old monetary system, in which silver was the standard, 12 Pfennig =1 Schilling = ^ltd.\ and 16 Schillings =1 Mark = i3>i^. Bremen. — Old system: — 5 Schmaren= i Groot = "/aorfl; and 71 Groots = I Rix-dollar=3J. z^li^- The Rix-dollar, or Thaler, was a money of account Luhec. — ^The old system was the same as that of Hamburg, and, in addition, 3 Marks = i Thaler = 3 j. 4^/. 192 MONEY. AUSTRIA.— Money. J. d, I Kreutzer (copper) o '/j 4 Kreutzers (do.) o 4/^ 10 Kreutzers (silver) o 2^ 20 Kreutzers i^o.) o 4^^ ^ Florin (do.) o sH' 1 Florin (do.) i 11^ 2 Florins (do.) 3 11}^ 4 Florin piece (gold) 7 11 8 Florin piece (do.) 15 10 100 Kreutzers make i Florin. The 4-florin gold piece weighs 49.92 grains, and the standard of fineness is 90 per cent of gold. RUSSIA. — Money. I Copeck 1 00 Copecks I Silver Rouble . s. d. = .38 — 3 2 The copper coins are pieces of ^, ^, i, 2, 3, 5 Copecks. The silver coins are pieces of 5, 10, 15, 20, 25 Copecks, the Half Rouble, and the Rouble; the gold coins are the Three-rouble piece, the Half Imperial of five Roubles, and the Imperial of 10 Roubles. The 5-rouble gold piece weighs 1 01 grains, and the standard of fineness is 91^ per cent of gold. Paper currency: — i, 3, 5, 10, 25, 50, 100 Roubles. HOLLAND.— Money. s, d, I Cent = o '/j 100 Cents I Guilder or Florin = i 8 BELGIUM.— Money. The monetary system is exactly the same as that of France. DENMARK.— Money. s, d, I Skilling = o .2745 16 Skillings i Mark = o 4.392 96 Skillings, or 6 Marks i Rigsdaler, or Daler = 22 7/ao SWEDEN. — Money. s, d, I Ore = o .133 100 Ore I Riksdaler = i i^ NORWAY.— Money. s, d. I Skilling = o .444 24 Skillingen i Ort or Mark = o lo^ 5 Ort I Species-Daler = 4 $)i SWITZERLAND, SPAIN, ETC I93 SWITZERLAND.— Money. The monetary system of Switzerland is the same as that of France. The Centime is called a Rappe. SPAIN. — Money. d. I Centimo = 95 100 Centimos i Peseta = i franc, or 9)^ The bronze coins are pieces of i, 2, 5, and 10 centimos. The silver coins are pieces of 20 and 25 centimos, and i, 2, and 5 pesetas. The gold coins are pieces of 5, 10, 20, 25, 50, and 100 pesetas. The piece of 5 pesetas is y, iij4^., English value. The 25 peseta piece is 19J. 9^^., English value. The old monetary system was based on the Real-Vellon, 2j^^. English value; it was the 20th part of the Silver Hard Dollar^ 4^. 2//. English value, and of the Gold Dollar or Coronilla. The Duro was identical with the American Dollar. PORTUGAL.— Money. The unit of account is the Rei, of which 185^ Reis make i penny; and 4500 Reis make i sovereign. The Milreis is 1000 Reis, ^r. syid, English value. The Corda is the heaviest gold coin, of 10,000 Reis, JQ2, ^r. 5^//. English value, and weighs 17.735 grammes. ITALY. — Money. d, I Centime = .95 100 Centimes i Lira = i franc, or 9^ Copper coins are pieces of i, 3, and 5 Centimes; silver coins, 20 and 50 Centimes, and i, 2, and 5 Lire; gold coins, 5, lo, 20, 50, and 100 Lire. lliese coins are the same in weight and fineness as the coins of France. TURKEY. — Money. s, d, I Para = o 7,8.5 40 Paras i Piastre = o 2.16 100 Piastres i Medjidie, or Lira Turca = 18 o The Piastre is roughly taken equal to 2d. sterling. GREECE AND ;ONIAN ISLANDS.— MONEY. icx) Lepta i Drachma = i franc, or 9 J^//. The currency of Greece is the same as that of France. In the Ionian Islands, whilst they were under British protection (1830- 1864), accounts were kept by some persons in Dollars, of 100 Oboli = 4f. 2//.; by others in Pounds, of 20 shillings, of 12 pence, Ionian currency; the Ionian Pound being equal to 20s. g,6d sterling. By other persons accounts were kept in Piastres of 40 Paras =^ 2 */^^, 13 194 MONEY. MALTA. — Money. I Grano 20 Grani i Taro 12 Tari I Scudo Or, 60 Piccioli I Carlino 9 Carlini i Taro 12 Tari i Scudo s. d. v» I?^ I 8 .185 x?^ I 8 British money is in general circulation. The Sovereign = 12 Scudi; the Shilling = 7 Tari 4 Gram. EGYPT. — Money. £ J. //. I Para = 00 .0615 40 Paras i Piastre (Tariff) = o o 2.461 100 Piastres i Egyptian Guinea = i o 6.84 5 Egyptian Guineas... I Kees, or Purse = 52 10.2 1000 Purses I Khuzneh, or Treasury = 5142 10 o 97.22 Piastres i English Sovereign. The Egyptian guinea weighs 132 grains, and the standard of fineness is 87^ per cent of gold. Two piastres (current) are equal to one piastre (tariff). MOROCCO.— Money. I Flue 24 Flues I Blankeel = 4 Blankeels i Ounce = 10 Ounces i Mitkul = TUNIS. — Money. I Fel =0 3S/,88 3 Fels I Karub = o 3S/^ 16 Karubs i Piastre = o S ^/e ARABIA. — Money. s. d 80 Caveers i Piastre or Mocha Dollar =35 CAPE OF GOOD HOPE. — MoNEY. Public accounts are kept in English money; but private accounts are often kept in the old denominations, as foUows : — I Stiver = 6 Stivers i Schilling = 8 Schilling i Rix-doUar = The Guilder is equal to 6d. s. //. 37/960 3V40 3-7 3 I s. d 3/8 ^H I 6 INDIAN EMPIRE, CHINA, ETC 1 95 I14DIAN EMPIRE.— Money. Throughout India, accounts are kept in the following moneys: — s, d. I Pie = o o^ nominal value. 12 Pies I Anna. = o ij4 do. 16 Annas i Rupee = 20 do. The intrinsic value of the Rupee is is. io}id.; it weighs 180 grains. The English Sovereign is equal to 10 Rupees 4 Annas. I Lac of Rupees = 100,000 rupees = ^10,000. I Crore of Rupees =100 lacs = ;;^i, 000,000. In Ceylon, the Rupee is divided into 100 Cents. The gold coin, Mohur, is equal to 15 rupees; it weighs 180 grains, and the standard fineness is 91.65 per cent of gold. CHINA.— Money. J. a. I Cash (Le) = o 7/,oo 10 Cash I Candajreen (Fun) = o y/,© 10 Candareens i Mace (Tsien) =0 7 10 Mace iTael(Leang) = 5 10 COCHIN-CHINA.— Money. . s. a, I Sapek, or Dong, or Cash = o «/i8 60 Sapeks i Mas, or Mottien = o 3^ 10 Mas. I Quan, or String. = 2 9^ PERSIA.— Money. . s, a. I Dinar = o '/a© 50 Dinars i Shahi =0 ^ 20 Shahis i Keran = o iij^ 10 Kerans i Toman = 9 3J6 JAPAN.— Money. 10 Rin. I Sen = J^ 100 Sen I Yen =42 There are gold coins of the value of i, 2 and 5 yen, with a standard fineness of 90 per cent The 5-yen piece weighs 128.6 grains. The silver yen weighs 416 grains, with the same standard of fineness. JAVA. — Money. The money account of Java is the same as that of Holland. UNITED STATES OF AMERICA.— MoNEY. s. d, I Cent =0 }i 10 Cents I Dime.... =05 100 Cents I Dollar. =42 196 MONEY. CANADA.— BRITISH NORTH AMERICA.— MoNEY. s. d. I Mil = o Vao sterling. 10 Mils I Cent = o J^ do. 100 Cents I Dollar = 4 i}( do. 4 Dollars --= 20 o currency. Or, I Penny currency = o ^ sterling. 12 Pence i Shilling do = o 94/^ do. 20 Shillings i Pound do = 16 5^ do. The Dollar of Nova Scotia, New Brunswick, and Newfoundland, is equal to 4J. 2d. sterling. In the Bermudas, accounts are kept in sterling money. MEXICO.— Money. Accounts are kept in dollars of 100 cents. The dollar is equal to 4s, 2d. sterling. CENTRAL AMERICA AND WEST INDIES.— MoNEY. WEST INDIES (British). Accounts are kept in English money; and sometimes in dollars and cents. I dollar = 4r. 2d, CUBA. — Money. The moneys of various nations were in circulation before the current war (1875). But the principal silver currency was the 10 cent and 5 cent pieces of the United States. The gold currency consists of the Ounce, of the value of 16 dollars, }4 ounce, jounce, }i ounce. GUATEMALA, HONDURAS, COSTA RICA. The mone)rs of account are the same as those of Mexico. ST. DOMINGO. Accounts are kept in current dollars (called Gourde) and cents. The cent= Vsa^'i ^^d ^00 cents = i dollar = 3^//. SOUTH AMERICA. — MoNEY. COLOMBIA, VENEZUELA, ECUADOR. The moneys of account are, the Centavo= J^^.; and 100 Centavos = I Peso = 4J'. 2d, GUIANA. In British Guiana the dollar of 4s, 2d, is used, divided into 100 cents. In French Guiana, French money is used. In Dutch Guiana, the money of Holland is used. BRAZIL, PERU, ETC. 1 97 BRAZIL. — Money. s. d. I Rei - o «Vioo looo Reis '. I Milreis - 2 3 PERU. — Money. s. d, I Centesimo =0 .37 1 00 Centesimos i Dollar, or Peso =31 CHILI. — Money. J. d/ I Centavo =0 .45 100 Centavos i Dollar, or Peso =39 BOLIVIA. I Centena =0 .37 100 Centenas i Dollar =31 ARGENTINE CONFEDERATION. I Centesimo =0 .25 100 Centesimos i Dollar, or Patercon =21 URUGUAY. I Centime =0 oj4 100 Centimes i Dollar =42 PARAGUAY. I Centena =0 .37 100 Centenas i Dollar =31 AUSTRALASIA. Accounts are kept in pounds, shillings, and pence sterling. WEIGHT AND SPECIFIC GRAVITY. The specific gravity, or specific weight of a body, is the ratio which the weight of the body bears to the weight of another body of equal volume adopted as a standard for comparison of the weights of bodies. For solids and liquids, pure water at the mean temperature 62** F., is adopted as the standard body for comparative weight. For gases, dry air at 32° F., and under one atmosphere of pressure, or 14.7 lbs. per square inch, is the body with which they are compared. The specific gravity of bodies is found by weighing them in and out of water, according to the following rules. Rule i. — To find the specific gravity of a solid body heavier than water. Weigh it in pure water at 62® F., and divide its weight out of water by the loss of weight in the water. The quotient is the specific gravity. Note, — The loss of weight in water is the difference of the weight in air and the weight in water, and it is equal to the weight of the quantity of water displaced, which is equal in volume to the body. Rule 2. — To find the specific gravity of a solid body lighter than water. Load it so as to sink it in pure water at 62'' F., and weigh it and the load together, out of water, and in water; weigh the load separately in and out of water; deduct the loss of weight of the load singly from that of the combined body and load ; the remainder is the loss of weight of the body singly, by which its weight out of water is to be divided. The quotient is the specific gravity. Rule 3. — To find the specific gravity of a solid body which is soluble in water. Weigh it in a liquid in which it is not soluble; divide the weight out of the liquid by the loss of weight in the liquid, and multiply by the specific gravity of the liquid. The product is the specific gravity of the body. Rule 4. — To find the specific gravity of a liquid. Weigh a sblid body in the liquid and in water, as well as in the air, and divide the loss of weight in the liquid by the loss of weight in water. The quotient is the specific gravity. Rule 5. — To find the weight of a body when the specific gravity is given. Multiply the specific gravity by MULTIPLIES. WEIGHT OP 62.355 (t^^ weight in pounds of a cubic foot of piure water at 62° F.) = i cubic foot, in lbs. 1683.60 =1 cubic yard, in lbs. 15.0 =1 „ incwts. .75 ^i „ intons. WEIGHT AND SPECIFIC GRAVITY. 1 99 Note, — ^As one cubic foot of water at 62° F. weighs about 1000 ounces (exactly 997.68 ounces), the weight in ounces of a cubic foot of any other substance will represent, approximately, its specific gravity, supposing water =1000. If the last three places of figures be pointed off as decimals, the result will be the specific gravity approximately, water being = i. In France, the standard temperature for comparison of the density of bodies, and the determination of their specific gravities, is that of the maximum density of water, — about 4° C, or 39°.! F., for solid bodies; and 32' F., or 0° C, for gases and vapours, under one atmosphere or .76 centi- metres of mercury. In practice, it is usual to adopt the cubic decimetre or litre as the unit of volume, since the cubic decimetre of distilled water, at 4* C. weighs, by the definition, i kilogramme. Consequently the specific gravity of a body is expressed by the weight in kilogrammes of a cubic decimetre of that body. The densities of the metals vary greatly. Potassium and one or two others are lighter than water. Platinum is more than twenty times as heavy. Lead is over eleven times as heavy; and the majority of the useful metals are from seven to eight times as heavy as water. Stones for building or other purposes vary in weight within much narrower limits than metals. With one exception, they vary from basalt and granite, which are three times the weight of water, to volcanic scoriae which are lighter than water. The exception referred to is barytes, which is con- spicuously the heaviest stone, being 4j^ times as heavy as water. The sulphate of baryta is known as heavy spar. Amongst other solids, flint-glass has three times the weight of water; clay and sand, twice as much; coal averages one and a half times the weight of water; and coke from one to one and a half times. Camphor has about the same weight as water. Of the precious stones, zircon is the heaviest, having four and a half times the weight of water; garnet is four times as heavy, diamond three and a half times as heavy, and opal, the lightest of all, has just twice the weight of water. Peat varies in weight from one-fifth to a little more than the weight of water. The heaviest wood is that of the pomegranate, which has one and a third times the weight of water. English oak is nearly as heavy as water, and heart of oak is heavier; the densest teak has about the same weight as water; mahogany averages about three-fourths, elm over a half, pine from a half to three-fourths, and cork one-fourth of the weight of water. Of the colonial woods, the average of 22 woods of British Guiana weighs 74 per cent, of the weight of water; of 36 woods of Jamaica, 83 per cent.; and of 18 woods of New South Wales, 96 per cent Wood-charcoal in powder averages one and a half times the weight of water; in pieces heaped, it averages only two-fifths. Gunpowder has about twice the weight of water. Of animal substances, pearls weigh heaviest, two and three-quarter times the weight of water; ivory and bone twice, and fat over nine-tenths the weight of water. Of vegetable substances, cotton weighs about twice as much as water; gutta-percha and caoutchouc nearly th^ §apa^ weight ^ waiter, 200 WEIGHT AND SPECIFIC GRAVITY Mercury, the heaviest liquid at ordinary temperatures, has over thirteen and a half times the weight of water; and bromine nearly three times the weight The water of the Dead Sea is a fourth heavier, and ordinary sea- water two and a half per cent, heavier than water; whilst olive-oil is about one-tenth lighter, and pure alcohol and wood-spirit a fifth lighter than water. Turning to gaseous bodies, water at 62** F. has 772.4 times the weight of air at 32° F., under a pressure of one atmosphere; and the specific gravity of air at 32° F. is .001293, that of water at 62*^ F. being = i. Oxygen gas weighs a tenth more than air, gaseous steam weighs only five-eighths of air, and hydrogen, the most perfect type of gaseity, has only seven per cent, of the weight of air. Water has upwards of 11,000 times the weight of hydrogen. One pound of air at 62° F. has the same volume as a ton of quartz. The following Tables, Nos. 65 to 69, contain the weights and specific gravities of solids, liquids, and gases and vapours. The specific gravities have been derived from the works of Rankine, Ure, Wilson, Claudel, and Peclet, Delabfeche* and Playfair, Fowke, and others whose names are men- tioned in the body of the tables. Columns containing the bulks of bodies have been added to the tables. The specific gravity of alloys does not usually follow the ratios of those of their constituents; it is sometimes greater and sometimes less than the mean of these. Ure gives the specific gravities of some alloys of copper, tin, zinc, and lead, examined by Crookewitt. The following are the specific gravities of the alloys, as ascertained by Crookewitt; and, for the purpose of comparison, they are preceded by the specific gravities of the particular samples of the elementary metals employed. SPECIFIC GRAVITY. Copper 8.794 Tin 7.305 Zinc 6.860 Lead ii-3S4 Alloys: — Copper 2, tin 5 7«6S2 Copper I, tin i 8.072 Copper 2, tin i 8.512 Copper 3, zinc 5 7.939 Copper 3, zinc 2 8.224 Copper 2, zinc i 8.392 Copper 2, lead 3 10-753 Copper I, lead i io-375 Tin I, zinc 2 7.096 Tin I, zinc I 7-ii5 Tin 3, zinc i 7.235 Tin I, lead 2 9*965 Tin I, lead I 9*394 Tin 2, lead I 9.025 The following binary alloys have, on the one side, a density greater than the mean density of their constituents; and, on the other side, a density less than the mean density of the constituents. OF METALS AND ALLOYS. 20 1 Alloys liaving a density greater than the mean. Gold and zinc. Gold and tin. Gold and bismuth. Gold and antimony. Gold and cobalt Silver and zinc. Silver and lead. Silver and tin. Silver and bismuth. Silver and antimony. Copper and zinc. Copper and tin. Copper and palladium. Copper and bismuth. Lead and antimony. Platinum and molybdenum. Palladium and bismuth. Alloys having a density less than the mean. Gold and silver. Gold and iron. Gold and lead. Gold and copper. Gold and iridium. Gold and nickel. Silver and copper. Iron and bismuth. Iron and antimony. Iron and lead. Tin and lead. Tin and palladium. Tin and antimony. Nickel and arsenic. Zinc and antimony. 202 VOLUME, WEIGHT, AND SPECIFIC GRAVITY TABLE No. 65.— VOLUME, WEIGHT, AND SPECIFIC GRAVITY OF SOLID BODIES. FAMILIAR METALS. Platinum Gold Mercury, fluid Lead, milled sheet Do. wire : Silver Bismuth Copper, sheet Do. hammered Do. wire Bronze: — 84 copper, 16 tin, gun metal 83 » 17 » « 81 » 19 i> « 79 „ 21 „ mill-bearmgs 35 » 65 „ small bells 21 » 79 V » 15 „ 85 „ speculum metal... Nickel, hammered Do. cast Brass: — cast 75 copper, 25 zinc, sheet 66 „ 34 n yellow 60 „ 40 „ Muntz's metal, ... Brass, wire Manganese Steel : — Least and greatest density Homogeneous metal Blistered steel Crucible steel Do. average '. Cast steel, Do. average Bessemer steel Do. average M ean for ordinary calculations Iron, wrought : — Least and greatest density... Common bar Puddled slab Various — Irons tested by Mr. Kirkaldy Do. average Common Tails Do. average Yorkshire iron bar Lowmoor plates, i|i to 3 ins. thick.... Beale's rolled iron Pure iron (exceptional), by electro- ) deposit (Dr. Percy) ) Mean, for ordinary calculations Weight of one cubic foot pounds. .. 1342 ... I2CO 849 ... 712 704 ... 655 617 ... 549 .. 556 ... 554 534 ... 528 .. 520 ..• 544 503 ... 461 465 ... 541 516 ... 505 527 ... 518 511 533 499 ... 435 to 493 493 ... 488 ..488 to 490... 489 489 to 489.5 489.3 ..489 to 490... 489.6 489.6 ... 466 to 487 471 ... 460.5 to 474 ..468 to 486... 477 ..466 to 476... 470 484 ... 487 476 ... Specific Gfavity. Watcr= I. 508 480 ... 21.522 19.245 . ... 13.596 II.418 ... 11.282 10.505 9.90 8.805 ... 8.917 8.880 ... 8.56 8.46 8.46 8.73 ... 8.06 7.39 ... 7.45 8.67 ... 8.28 8.10 ... 8.45 8.30 8.20 8.548 ,.. 8.00 7.729 to 7.904 .. 7.904 7.823 7.825 to 7.859 7.842 7.844 to 7.851 7.848 7.844 to 7.857 7.852 .. 7.852 7.47 to 7.808 .. 7.55 7.53 to 7.60 .. 7.5 to 7.8 7.65 7.47 to 7.64 7.54 .. 7.758 7.808 .. 7.632 8.140 7.698 OF SOLID BODIES. 203 Familiar Metals {continued). Iron, cast: — Least and greatest density White Gray ; Eglinton hot-blast, ist melting... 2d do. . . . 14th do. ... Rennie Mallett Mean, for ordinary calculations.. Tin Zinc, sheet Do. cast Antimony Aluminium, wrought Do. cast Magnesium OTHER METALS. Iridium Uranium Tungsten Thallium Palladium..... Rhodium Osmium Cadmium Molybdenum Ruthenium... Cobalt Tellurium Chromium.... Arsenic Titanium Strontium Glucinum Calcium Rubidiimi Sodium Potassium...., Lithium Weisht of one cubic fooL pounds. 378.25 to 467.66 468 449 - 435 435 ... 470 435 to 444... 442 450 ... 462 449 ..- 428 418 167 160 108.5 1 165.0 1 147.0 1097.0 742.6 735.8 660.9 623.6 542.5 537.5 536.2 530.0 381.0 374.1 361.5 330.5 158.4 131.0 98.5 94.8 60.5 53-6 37.0 Specific Gravity. Water = i. 6.900 to 7.500 7.50 7.20 6.969 6.970 7.530 6.977 to 7. 1 13 7.094 ... 7.217 7.409 7.20 6.86 6.71 2.67 2.56 1.74 PRECIOUS STONES. Zircon Garnet Malachite Sapphire Emerald Do. Aqua marine.. Amethyst Ruby Diamond. Specific Gravity. .. 4.50 3.60 to 4.20 .. 4.01 3.98 .. 3.95 2.73 .. 3.92 3.95 3.50 to 3.53 Diamond, Pure Boart Topaz Tourmaline Lapis lazuli Turquoise Jasper, Onyx, Agate.... Beryl.: Opal 18.68 18.40 17.60 11.91 11.80 1 0.60 10.00 8.70 8.62 8.60 8.50 6.1 1 6.00 5.80 5.30 2.54 2.10 1.58 1.52 0.97 0.86 0.59 Specific Gravity. ... 3.52 3.50 ... 3.50 3.07 ... 2.96 2.84 2.6 to 2.7 2.68 2.09 204 VOLUME, WEIGHT, AND SPECIFIC GRAVITY STONES. Specular, or red iron ore Magnetic iron ore Brown iron ore Spathic iron ore , Clydesdale iron ores , Barytes Basalt Mica Limestone, Magnesian Do. Carboniferous Marble: — Paros African Pyrenean , Egyptian, green French Florentine, Sienna Trap, touchstone Granite, Sienite, gneiss Do. Gray Porphyry Alabaster, Calcareous Do. Gypseous Chalk, Air-dried Slate Serpentine Potter's Stone Schist, Slate Do. Rough Lava, Vesuvian Talc, Steatite Rock Crystal Quartz Do. Crystalline Do. for paving Do. porous, for millstones Do. flaky, for do Flint Felspar Gypsum Lias Graphite Sandstone Tufa, volcanic Scoria, do Cubic feet to one ton, solid. cubic feet. 6.84 • * 7.05 9.16 .. 9.38 .. 11.76 .. 8.07 14-7 to 12.0 14.0 to 12.3 ,. 12.6 .. 13.3 .. 12.7 .. 12.8 .. 13.2 .. 13.2 .. 13.2 .. 13.5 .. 13.6 .. 14.3 .. 13.2 15.2 to 12.1 12.8 to 11.8 13-5 to 13.1 .. 13.0 .. 15.6 14.9 to 14. 1 13.8 to 12.6 .. 12.8 .. 12.8 .. 12.8 19.9 to 12.9 21.0 to 12.8 13.3 13.6 13.8 to 13.3 13.6 .. 14.4 .. 28.5 I4.I 13-7 .. 13.8 .. 15.6 16.0 to 14.7 16.3 17.3 to 14.3 29,7 to 26.1 .. 43.3 •. Weight of one cubic foot, solid. pounds. ••• 3274 317.6 ... 244.6 .. 238.8 190.5 .. 277.5 152.8 to 187.1 160.3 to 182.7 ... 178.3 .. 168.0 ... I77.I .. 174.6 170.2 .. 170.2 169.6 .. 166.5 165.2 .. 157.1 169.6 .. 147.1 to 184.6 174.6 to 190.8 166.5 to 171.5 ... I72.I 144.0 ...150 to 159.. 162.1 to 177.7 ... 175.2 .. 174.6 ... 174.6 .. 112.8 to 173.3 106.6 to 175.2 168.4 165.2 162.8 to 169.0 165.2 ... 155.9 .. 78.6 ... 159.0 .. 164.0 ... I62.I .. 1434 140.3 to 152.8 137.2 129.7 to 1 57. 1 75.4 to 86.0 ... 51.7 .. Specific Gravity. • B I • • Water = I. 5.251 5.094 ... 3.922 3.829 3.055103.380 4-45 2.45 to 3.00 2.57 to 2.93 2.86 2.69 . .. 2.04 2.80 ... 2.73 2.73 • • i 2.67 2.65 2.52 ... 2.72 2.36 to 2.96 2.80 to 3.06 2.67 to 2.75 ... 2.76 2.31 2.46 to 2.55 2.60 to 2.85 ... 2.81 2.80 2.80 1. 81 to 2.78 1.71 to 2.81 2.70 2.65 2.61 to 2.71 2.65 ... 2.50 1.26 ... 2.55 2.63 2.60 2.30 2.25 to 245 2.20 2.08 to 2.52 I.2I to 1.38 •83 1 \ OF SOLID BODIES. 201; SUNDRY MINERAL SUBSTANCES.' Glass:— Flint Green Plate Crown St. Gobain Common, with base of potash Fine, do. do. Common, with base of soda... Fine, do. do. ... Soluble Porcelain :— China Sevres Portland Cement Concrete : — P. cement i, and shingle 10 P. cement, rubble, and sand P. cement i, and sand 2 Roman cement i,and sand 2 Mortar. Brick Brickwork Masonry, Rubble MarL Do. very tough Potash Sulphur TUes Rock Salt Conunon Salt, as a solid Clay Sand, pure earthy Earth :— Potter's Argillaceous Light vegetable Mud Materials in the bed of the Clyde :- Fine sand and a few pebbles, laid in a box, loose, not pressed, nearly dry Pressed Mud at Whiteinch, dry, and ] firmly packed, containing > very fine sand and mica ) Wet mud, rather compact and firm,well pressed into the box Wet, fine, sharp gravel, well pressed Wet, running mud Sharp dry sand deposit, in ^ harbour ) Port-Glasgow bank (sand), wet, ) pressed into a box \ ] Cubic feet to one ton, solid. cubic feet. Weight of one cubic foot, solid. 28.7 to 23.8 .. 16.1 16.6 to 16.0 .. 17.6 .. 18.7 20.6 1 8. 1 to 16.0 20.4 to 19.5 19.4 to 15.6 22.4 to 18.9 .. 17.1 18.0 .. 18.0 .. 17. 1 to 15.9 .. 18.7 .. 18.7 .. 18.9 .. 21. 1 .. 18.9 .. 22.4 .. 25.7 .. 22. pounds. 187.0 168.4 168.4 155-9 155.3 153-4 152.8 152.8 1 52. 1 77.9 148.4 139-7 78 to 94 26 24 23 19 18 18.1 24-3 18.6 .. 139 135 to 140 127 120 109 124.7 to 135.3 ..no to 115. 1 1 5.3 to 143.4 99.8 to 1 18.5 146 .. 131 124-7 124.7 •• 131 to 140.7 .. II9.7 .. 1 19.7 .. II8.5 .. 106.0 .. II8.5 .. 99.8 87.3 .. 1 01. 6 Specific Gravity. 87 92 97 115 124 122X 92 120.5 Water = i. 3.00 2.70 2.70 2.50 2.49 2.46 2.45 2.45 2.44 1.25 2.38 2.24 1.25 to 1. 51 .. 2.23 2.17 to 2.25 .. 2.04 1.92 .. 1.75 2.00 to 2.17 1.76 to 1.84 1.85 to 2.30 1.60 to 1.90 2.34 2.10 2.00 2.00 2.IOOtO 2.257 1.92 1.92 1.90 1.70 1.90 1.60 1.40 1.63 ... 1.39 1.48 ... 1.56 ... 1.95 1.99 1.97 ... 1.48 ... 1.93 206 VOLUME, WEIGHT, AND SPECIFIC GRAVITY Mineral Substances {continued). Materials in the bed of the Qyde ; — Sand opposite Erskine House, ) wet, pressed J Alluvial earth, pressed Do. do. loose Plaster: — 24 hours after using 2 months after using ... Coal, Anthracite (see Sect. Coal) Bituminous do. do. Boghead (cannel) do. do. Coke Phosphorus Alum Camphor Meltmg Ice Cubic feet to one ton, solid. cubic feet. 19.3 24 .. 33 22.6 .. 25.7 .. 26.2 to 22.6 30 to 28.1 30 39 to 21.6 20.3 20.9 .. 36.3 •• 39 Weight of one cubic foot, solid. pounds. 116 93 67 99.2 .. 87.3 .. 85.4 to 99.1 74.8 to 81.7 74.8 57.4 to 103.5 1 10.4 .. 107.2 .. 61.7 .. 57.5 .. Specific Gravity. Waters X. .. 1.86 1.49 .. 1.08 1.59 1.40 1.37 to 1.59 1.20 to 1. 31 1.20 .92 to 1.66 1.77 .. 1.72 •99 .922 • • • COALS. Delabeche and Playfair,) Welsh: — ^Anthracite Porth Mawr (highest) Llynvi (one of the lowest) Average of 37 samples Newcastle: — Hedley's Hartley (highest) ... Original Hartley (one of the lowest) Average of 18 samples Derbyshire and Yorkshire: — Elsecar Butterley Stavely Loscoe, soft Average of 7 samples Lancashire: — Laffack Bushy Park (highest) Cannel, Wigan (lowest) Average of 28 samples Scotch : — Grangemouth (h ighest) Wallsend Elgin Average of 8 samples Irish : — Slievardagh Anthracite Warlich's artificial fuel Cubic feet in a ton. Heaped. cubic feet. 38.4 42.0 42.0 42.7 431 45.6 45-3 47.4 47-3 44.9 48.8 47.4 42.6 46.4 45.2 40.1 41.0 42.0 35.7 32.4 Weight of one cubic foot. SoUd. pounds. 85.4 86.7 80.3 82.3 81.8 78.0 78.3 80.8 79.8 79.8 79.6 79.6 84.1 76.8 79-4 80.5 74.8 78.6 99.6 72.2 Heaped. pounds. 58.3 53-3 53.3 53.1 52.0 49.1 49.8 47.2 47.4 49-9 45.9 45-9 52.6 48.3 49.7 54.3 54.6 50.0 62.8 69.6 Specific Gravity. Water = i. .37 .39 .28 •315 •31 .256 .296 .28 .27 .285 .292 •35 .23 .273 .29 .20 .259 .59 •15 OF SOLID BODIES. 207 PEAT. {Dr, Sullivan,) Irish peat (comprising an average amount of water from 20 to 25 per cent) : — Lightest upper moss peat ... Average lignt moss peat Average brown peat Compact black peat Mean of five samples {Another observation,) Average upper brown peat .. Moderately compact lower ) brown turf ) Mean of two classes Condensed peat {Kane and Sullivan,) Excessively light, spongy ) surface peat ) Light surface peat Rather dense peat Very dense dark brown peat Very dense blackish brown ) compact peat J Exceedingly dense jet black \ ^P«at ( Exceedingly dense, dark, \ blackish brown peat \ (/CarmarscA.) Turfy peat, Hanover Fibrous peat, do Earthy peat, do Pitchy peat, do Cubic feet per ton, stalked. cubic feet. ...369.60. 254.20 ...147.00. 131.28 ... 99.36. 200.29 . . . I oo.O . . . 155-5 ...141.75. 5 1.2 to 40.0 Weight of one cubic foot, stalked. pounds. 6.06 8.81 15-13 17.06 22.54 11.18 .. 11.92 14.40 ,. 15.80 43.75 to 56.8 Weiffht of one cubic foot, solid. pounds. 62.5 to 81. 1 13.7 to 21.0 20.9 to 25.3 29.7 to 41.7 40.5 to 44.5 45.1 to 61.3 53.2 to 61.8 . . . 66.0 . . . 6.9 to 16.2 1 5.0 to 41.8 2 5.6 to 56.1 38.7 to 64.2 Specific Gravity. Water = i. i.o to 1.3 .2 19 to .337 .335 to .405 .476 to .669 .65010.713 .72410.983 .72510.991 ... 1.058 .II to .26 .24 to .67 .41 to .90 .62 to 1.03 FUEL IN FRANCE. {ClaudeL) Pure Graphite. Anthracite Rich coaly with ^ long flame Dry coal, with a long flame Rich and hard coal Smithy coaL Lignite Do. bituminous Do. imperfect «Jayet»...r! Bitumen, red Do. black Do. brown Asphalte Weisht of one cubic foot pounds. .. 145-3 •• 83.5 to 91.0 79.8 to 84.8 84.8 .. 82.3 .. 79.8 to 81. 1 77.9 to 84.2 72.3 to 74.8 68.6 to 74.2 81.7 .. 72.3 .. 66.7 ... 51.7 .. 66.1 Specific Gravity. Water = x. .. 2.33 1.34 to 1.46 1.28 to 1.36 1.36 .. 1.32 1.28 to 1.30 1.25 to, 1.35 1. 16 to 1.20 1. 10 to 1. 19 .. 1. 16 1.07 .. 0.83 1.06 2o8 VOLUME, WEIGHT, AND SPECIFIC GRAVITY WOODS. Pomegranate Boxwood Do. of Holland Do. of France Lignum vitae Ebony Do. Green Do. Black Oak, Heart of. Do. English Do. European Do. American, Red Lancewood Rosewood Satin-wood Walnut, Green Do. Brown Laburnum Hawthorn Mulberry Plum-tree Teak, African » Mahogany, Spanish Do. St. Domingo Do. Cuba Do. Honduras Beech Do. with 20 per cent, moisture. Do. cut one year Ash.... Weight of one cubic foot. Do. with 20 per cent, moisture Acacia Do. with 20 per cent, moisture. Holly Hornbeam Yew Birch Elm Do. Green Do. with 20 per cent, moisture Yoke-Elm do. do Rock-Elm Fir, Norway pine Do. Red pine Do. Spruce Do. Larch Do. White pine, English Do. do. Scotch Do. do. do. 20 per cent, moisture.. Do. Yellow pine Do. do. American American Pine-wood, in cord (heaped) Apple-tree pounds. .. 84.2 .. 64.8 .. 82.3 .. 56.7 40.5 to 82.9 70.5 .. 75.5 .. 74.2 .. 730 .. 58.0 43.0 to 61.7 54.2 41.8 to 63.0 64.2 59-9 •• 57.4 42.4 .. 57-4 56.7 .. 55.5 54.2 .. Specific Gravity. 53.0 .. 46.8 34-9 •• , 34.9 46.8 to 53.0 51.1 ... 41.2 .. 52.4 ... 43.7 .. 51.1 ... 44-9 •• 47.5 ... 47.5 •• 46.1 to 50.5 44.9 to 46. 1 34-3 ... 47.5 .. 44-9 ... 47.5 .. 50.0 4^* ' 29.9 to 43.7 29.9 to 43.7 31.18 to 39.9 •.. 34.3 •• 34.3 30.0 .. 41.2 ... 28.7 .. 21 ... 45.5 .. •• 1-35 1.04 1.32 0.91 .65 to 1.33 1. 13 1.21 1.19 .. 1.17 0-93 .69 to .99 .87 .67 to 1. 01 1.03 .. 0.96 0.92 .. 0.68 0.92 .. 0.91 0.89 .. 0.87 .98 .. 0.85 0.75 .. 0.56 0.56 0.75 to 0.85 0.82 0.66 a84 .. 0.70 0.82 .. 0.72 0.76 .. 0.76 0.74 to 0.81 0.72 to 0.74 0-55 .. 0.76 0.72 0.76 0.80 .. 0.74 0.48 to 0.70 0.48 to 0.70 0.50 to 0.64 .. 0.55 0.53 .. 0.49 0.66 .. 0.46 0.34 .. 0.73 OF SOLID BODIES. 209 1 Pear-tree....: Orange-tree % Olive-tree Maple Do. 20 per cent, moisture Service-tree. Cypress, cut one year Plane-tree Vine- tree Aspen-tree., .\lder-tree Do. 20 per cent, moisture Sycamore Cedar of Lebanon Bamboo Poplar. Do. White Do. 20 per cent, moisture Willow Cork. Elder pith INDIAN WOODS. {Berkley^ Northern Teak Southern Teak Jungle Teak Blackwood Khair Enroul Red Eyne Bibla Peon Kullum Hedoo. COLONIAL WOODS. Jamaica:— Black heart ebony Lignum vitae Small leaf. Neesberry bullet-tree Red bully-tree Iron wood Sweet wood Fustic Satin candlewood , Bastard cabbage bark White dogwood Black do Gynip Weiffht of one cubic fooL pounds. 45-5 44-3 •• 42.4 40.5 .. 41.8 41.8 .. 41.2 40.5 .. 37-4 374 .. 34.9 37.4 .. 36.8 30.6 to 35.5 19.5 to 24.9 .. 24.3 .. 20.0 to 31.8 29.9 .. 30.6 15.0 .. 4.74 55 48 41 56 11 63 68 56 39 41 39 40.5 74.2 .. to 73.0 73.0 .. 65.5 62.36 .. 61.7 60.5 .. 60.5 59-9 •• 58.6 58.6 .. 58.0 58.0 .. Specific Gravity. 073 .. 0.71 0.68 .. 0.65 0.67 .. 0.67 0.66 .. 0.65 0.60 0.60 0.56 0.60 0.59 0.49 to 0.57 0.31 to 0.40 •• 0.39 0.32 to 0.51 .. 0.48 0.49 .. 0.24 0.076 0.882 0.770 0.658 0.898 1. 171 I.0I4 I.09I 0.898 0.625 0.658 0.625 1. 19 0.65 to 1. 17 .. 1. 17 1.05 I.OO 0.99 .. 0.97 0.97 .. 0.96 0.94 .. 0.94 0-93 .. 0.93 14 210 VOLUME, WEIGHT, AND SPECIFIC GRAVITY Colonial Woods {continual). Jamaica {continued): — Wild mahogany Cashaw Wild orange Sweet do Bullet-tree (bastard) Tamarind Do. wild Prune Yellow Sanders Beech French Oak Broad Leaf Fiddle Wood Prickle Yellow Boxwood Locust-tree Lancewood Green Mahogany Yacca. Cedar Calabash Bitter Wood Blue Mahoe Average of 36 woods of Jamaica New South Wales:— Box of Ilwarra Do. Bastard Do. True, of Camden Mountain Ash Kakaralli Iron Bark Do. broad-leaved Woolly Butt Black Do Water Gum Blue Do Cog Wood Mahogany Do. swamp Gray Gum Stringy Bark Hickory Forest Swamp Oak Mean of 18 woods of New South Wales.. British Guiana :-- Sipiri, or Greenhcart Wallaba Brown Ebony Letter Wood Cuamara or Tonka Monkey Pot Mora Weiffht of one cubic foot. pounds. .. 57.4 .. 574 53.0 to 56.7 49-3 .. 56.1 .. 54.2 46.8 . 53.6 .. 53-6 .. 52.4 48.0 .. 44.3 .. 43-0 .. 43.0 .. 42.4 .. 42*4 *• 41.2 .. 39-3 .• 36.2 .. 34-9 •• 34.3 .. 337 .. 52.1 73-0 69.8 60.5 69.2 68.6 64.2 63.6 63.0 55.5 63.6 52.4 59-9 59.2 53.6 58.0 53.6 46.8 41.2 59-9 65.5 to 68.0 64.8 64*2 62.36 .. 61.7 .. 58.6 57.4 Spedfic Gtavity. 0.92 0.92 .85 to 0.91 0.79 0.90 0.87 0.75 0.86 0.86 0.84 0.77 0.77 0.71 0.69 0.69 0.68 0.68 0.66 0.63 0.58 0.56 0.55 0,54 0.835 . I.I7 1. 12 0.97 I.II 1. 10 1.03 1.02 l.OI 0.89 1. 00 0.84 0.96 0.95 0.86 0.93 a86 0.75 0.66 0.96 1.05 to 1.09 1.04 1.03 1. 00 .. 0.99 0.94 .. 0.92 J OF SOLID BODIES. 211 Colonial Woods {continued), British Guiana {continued) :— Ducaballi Cabacalli Kaiecri-balli Sirabuliballi Buhuradda Buckati Houbaballi Baracara. White Cedar Locust-tree. Cartan Purple Heart Bartaballi Crabwood : SilverbaJli Mean of 22 woods of British Guiana. ' WiUow Oak WOOD-CHARCOAL (as powder). {Ciaudel) Alder. Lime-tree Poplar Average of 5 charcoals, WOOD-CHARCOAL (in small pieces, heaped). Walnut Ash Beech Voke-elm. Apple-tree White Oak..... ^erry-tree.... Birch. EJm Yellow Pine.. Chestnut-tree. Poplar , Cedar {ClaudeL) Average of 13 charcoals Gunpowder WOOD-CHARCOAL (as made, heaped). Oak and Beech.. Birch Pine. Average. Weight of one cubic foot. pounds. 56.7 55.5 54.2 52.4 50.5 50.5 50.5 50.5 48.0 44.3 437 42.4 39-9 37.4 34-3 46.1 96.7 95-4 92.9 91.0 90.4 93.5 39-3 34-3 32.5 28.7 28.7 26.2 25.6 22.5 22.5 20.6 17.5 ... 15.6 15.0 ... 25.3 i09.itoii4.7 15 to 15.6 137 to 14.3 12.5 to 13.1 14 Specific Gravity. 0.91 0.89 0.87 0.84 0.81 0.81 0.81 0.81 0.77 0.71 0.70 0.68 0.64 0.60 0.55 0.74 1-55 1.53 1.49 1.46 145 1.50 .. 0.63 0.55 .. 0.52 0.46 .. 0.46 0.42 .. 0.41 0.36 .. 0.36 .. 0.28 0.25 .. 0.24 0.405 1.75 to 1.84 0.24 to 0.25 0.22 to 0.23 0.20 to 0.21 ,.. 0.225 212 WEIGHT AND VOLUME OF ANIMAL SUBSTANCES. {Claudel) Pearls Coral Ivory Bone Wool Tendon Cartilage Crystalline humour Human body Nerve Wax White of whalebone Butter Pork fat Mutton fat Animal charcoal, in heaps VEGETABLE SUBSTANCES. {Claudel) Cotton Flax Starch Fecula Gum — Myrrh Do. Dragon Do. Dragon's blood Do. Sandarac Do. Mastic Resin — Jalap Do. Guayacum Do. Benzoin Do. Colophany Amber, Opaque Do. Transparent Gutta-percha Caoutchouc Grain, Wheat, heaped Do. Barley, do Do. Oats, do Weight of one cubic foot. pounds. ... 169.6 ... 167.7 ... 1 19.7 ... 1 1 2.2 to 124.7 ... 100.4 ... 69.8 68.0 ... 66.7 64,9 ... 59.9 ... 58.7 ... 58.7 ... 58.7 574 50 to 52 121.6 1 1 1.6 95-4 93-5 84.8 82.3 74.8 68.0 66.7 76.1 74-8 68.0 66.7 68.0 67.3 60.5 58.0 46.7 36.6 31.2 Specific Gravity. .. 2.72 2.69 1.92 1.80 to 2.00 I.61 1. 12 1.09 1.08 .1.07 1.04 0.96 0.94 0.94 0.94 0.92 0.80 to 0.83 O, o, o, o o 95 79 53 50 36 32 20 09 07 22 20 09 07 09 08 97 93 75 59 50 VARIOUS SUBSTANCES. 213 TABLE No. 66.— WEIGHT AND VOLUME OF VARIOUS SUBSTANCES. {Tredgold,) SUBSTANCE. Lead (cast in pigs) Iron (cast in pigs) Limestone or marble (in blocks) Granite (Aberdeen, in blocks) ... Granite (Cornish, in blocks) Sandstone (in blocks) Portland stone (in blocks) Potter's clay Loam or strong soil Bath stone (in blocks) GraveL Sand. Bricks (common stocks, dry) Culm Water (river) Splint coal Oak (seasoned) Coal (Newcastle caking) Wheat Barley Red fir Hay (compact, old) Cubic feet per ton, in bulK. cubic feet. 4 6.25 13 13.5 14 16 17 17 18 18 21 23.5 24 36 36 39-5 43 45 47 59 59 280 Weight of one cubic toot, in bulk, lbs. 567 360 172 166 164 141 132 130 126 123.5 109 95 93 63 62.5 57 52 50 48 38 38 8 TABLE No. 67.— WEIGHT AND VOLUME OF GOODS CARRIED ON THE BOMBAY, BARODA, AND CENTRAL INDIA RAILWAY. By Colonel J. P. Kennedy, Consulting Engineer of the Railway. No. of kbuL I 2 3 4 I 7 Class I. CLASSIFICATION OF GOODS CONVEYED. Unpressed cotton . . . . Furniture Half-pressed cotton.. Cotton seeds Wool Fruit and vegetables. Eggs.... Averages Cubic feet per ton. Weight per ! cubic foot. cubic feet ... 224 ... 200 ... 186 ... 186" ... 140 ... 100 ... 90... lbs. ... 10 ... II ... 12 ... 12 ... 16 ... 22 ... 25 ... ...174... ... 13 ... Cubic feet per ton, in bulk (estimated). cubic feet. .. 280 250 ..233 ,..175 125 ... 113 ... 217 214 WEIGHT AND VOLUME OF GOODS. Goods conveyed over the Indian Railway {continued). No. of kind. 8 9 lO II 12 M 17 i8 19 20 21 22 Class 2. classification of goods conveyed. 23 ... 24 25 ... 26 27 ... 28 29 ... 30 31 .- 32 33 - 34 35 ..• 36 Class 3. 37 ..• 38 39 •• 40 41 ... 42 43 ... Qass 4. Grass Sundries Bagging .,., Commissariat stores Full-pressed cotton Flax and hemp Groceries Grains and seed Twist ^ Sugar Soap Firewood Salt Lime Dry Fruits Averages Jagree (Molasses) Kupas (Seed cotton) Mowra (flowers which produce spirit) Timber Ghee (clarified butter) Oil Piece goods Rape Beer and Spirits Coal Paper Tobacco Opium Machinery Averages Cutlery Potash Sand Colour Bricks Stone Metal Averages Averages of all classes Cubic feet per ton. culnc feet. 80 80 70 70 70 70 60 60 60 56 5> 51 50 Weight per cubic foot. 60 45 45 45 45 40 40 40 40 36 28 28 28 26 25 ... 41 20 20 20 18 17 15 5 ... II . . . 64.4 ' lbs. 28 28 32 32 32 32 37 37 37 40 40 40 44 44 45 37 50 50 50 ^A 5^ 5^ 5^ 56 62 80 80 80 86 90 54 112 112 112 124 132 148 443 354 Cubic feet per ton, m bulk (estimated). 203 ... cubic feet. ...ICX> ICO ... 87 87 ... 87 87 ... 75 75 ... 75 70 ... 70 70 ... 64 64 ... 63 ... 75 56 56 50 50 50 50 45 35 35 35 33 31 51 25 25 20 22 21 19 6X ... 14 80 J^oU, — The last column has been added bv the author; the quantities are calculated by adding one-fourth to the quantities in the third column, to give approximate estimate of the volume occupied in waggons by the goods, or the space required to load a ton of each kind. Sand, No. 39, lies solid in any situation. WEIGHT AND SPECIFIC GRAVITY OF LIQUIDS. 215 TABLE No. 68.-WEIGHT AND SPECIFIC GRAVITY OF LIQUIDS. LIQUIDS AT 33* F. Mercury Bromine. Sulphuric acid, maximum concentration.. Nitrous acid Chloroform. Water of the Dead Sea Nitric acid, of commerce Acetic acid, maximum concentration Milk. Sea water, ordinary Pure water (distilled) at 39°.! F Wine of Bordeaux. Do. Burgundy Oil, lintseed j Do. poppy 1 Do. rape-seed Do. whale Do. olive Do. turpentine '. Do. potato Petroleum Naphtha Ether, nitric Do. sulphurous Do. nitrous Do. acetic Do. hydrochloric Do. sulphuric Alcohol, proof spirit Do. pure Benzine Wood spirit Weight of one cubic foot. pounds. 848.7 185.I 1 14.9 96.8 95.5 774 76.2 67.4 64.3 64.05 62.425 62.1 61.9 58.7 58.1 57.4 57.4 57.1 54.3 51.2 54.9 69.3 67.4 55.6 55.6 54.3 44.9 57.4 49-3 53.1 49-9 Weight of one gallon. pounds. 136.0 . 29.7 18.4 , 15.5 15.3 . 12.4 12.2 . 10.8 10.3 , 10.3 1 0.0 . 9-9 9-9 • 9-4 9-3 . 9.2 9.2 . 8.7 . 8.2 8.8 . 8.5 II. I , 10.8 8.9 . 8.9 8.7 , 7.2 , 9.2 7.9 , 8.5 8.0 Specific Gravity. Water = i. ..13.596 2.966 .. 1.84 1.55 .. 1.53 1.24 .. 1.22 1.08 .. 1.03 1.026 .. I.OOO 0.994 .. 0.991 0.94 .. 0-93 0.92 .. 0.92 0.915 .. 0.87 0.82 .. 0.88 0.85 .. I. II 1.08 .. 0.89 0.89 .. 0.87 0.72 .. 0.92 0.79 .. 0.85 0.80 2l6 WEIGHT, ETC., OF GASES AND VAPOURS. TABLE No. 69.— WEIGHT AND SPECIFIC GRAVITY OF GASES AND VAPOURS. GASES AT 32* F. AND UKDBR ONE ATMOSPHERE OP PRESSURE. Vapour of mercury (ideal) Vapour of bromine Chloroform Vapour of turpentine Acetic ether Vapour of benzine Vapour of sulphuric ether .... Vapour of ether (.'*) Chlorine Sulphurous acid Alcohol Carbonic acid (actual) Do. (ideal) Oxygen Air Nitrogen Carbonic oxide Olefiant gas Gaseous steam Ammoniacal gas Light carburetted hydrogen .. Coal-gas (page 458) Hydrogen Volume of one pound weight. cubic feet. .. 1.776... 2.236 .. 2.337... 2.637 .. 4*075 ••• 4.598 .. 4.790 ... 4.777 .. 5.077 ... 5.513 . . 7.679 . . . 8.IOI .. 8.160 ... r 1.205 .. 12.307 ••• 12.727 .. 12.004 ••• 12.580 • .19.913 ... 21.017 ..22.412 ... 28.279 179.00 ... Weight of one cubic foot. in pounds. ...0.563 .. 0.447 ...0.428 ... 0.378 ...0.245 " 0.217 ...0.209 ... 0.206 ...0.197 ... O.1814 ...ai302 ... 0.12344 ...0.12259 0.089253 ...0.080728 0.078596 ...0.0781 ... 0.07808 .0.05022 0.04758 ...0.04462 0.03536 ...0.005594 m ounces. ..9.008 . 7.156 .6.846 . 6.042 .3-927 . 3.480 .3.340 . 3.302 .3.152 . 2.902 ...2.083 • 1.975 .. .1.961 . 1.428 .. .1.29165 1.258 ...1.250 . 1.249 ...0.8035 • 0.7613 ...0.7139.. 0.5658 ...0.0895 Specific Gravity. Air = X. .. 6.9740 5.5400 .. 5.3000 4.6978 .. 3.0400 2.6943 .. 2.5860 2.5563 .. 2.4400 2.2470 .. I.6130 1.5290 .. 1. 5180 1. 1056 .. I.OOOO 0.9736 .,0.9674. 0.9672 ..0.6220 0.5894. .0.5527 0.4381 ..0.0692 TABLES OF THE WEIGHT OF IRON AND OTHER METALS. Wrought Iron. — According to Table No. 65 of the Weight and Specific Gravity of Solids, the weight of a cubic foot of wrought iron varies, for various qualities, from 466 pounds to 487 pounds per cubic foot, and the average weight, taken for purposes of general calculation, is 480 pounds per cubic foot. This average weight is equivalent to a weight of 40 pounds per square foot, i inch in thickness — a convenient unit, which is usually employed in the development of tables of weights of iron for engineering and manufacturing purposes. The extremes of variation from this medium unit, extend from ^^ pound less, to about fi pound more than 40 pounds per square foot, or from 2.2 to 1.5 per cent, either way — a deviation, the extent of which is of little or no practical consequence, and which, at all events, is comprehended in the percentages allowed in the framing of estimates. The average weight of a cubic inch of ^vrought iron is i?^ =.277 pound, 1720 or one-tenth more than a quarter of a pound. For a round number, when cubic inches are dealt with, it may be, and is usually, taken as .28 pound, which is only four-fifths of i per cent, more than the medium weight, and corresponds to a weight of 483.84 pounds per cubic foot, or to 40.32 pounds per square foot, i inch thick, or to 10 pounds per lineal yard, I inch square. The volume of i pound of wTought iron is 3.6 cubic inches. Sied. — ^The weight of a cubic foot of steel varies from 435 pounds to 493 pounds per cubic foot, and the average weight is about 490 pounds per cubic foot. For convenience of calculation, the average weight is taken in the following tables, as 489.6 pounds per cubic foot, for which the specific weight is 1.02, when that of wrought iron = 1.00. The weight of a square foot, i inch thick, is 40.8 pounds; of a lineal yard, 10.2 pounds; and of a cubic inch, .283 pound. The volume of i pound of steel is 3.53 cubic inches. Cast Iron, — The weight of a cubic foot of cast iron varies from 378^^ pounds to 467^ pounds per cubic foot, and the average weight is taken as 450 ix)unds. The weight of a square foot, i inch thick is, therefore, 37.5 pounds; of a lineal yard, i inch square, 9.375 pounds; and a cubic inch, .26 pound. The specific weight is .9375. The volume of i pound of cast iron is 3.84 cubic inches. The following data, for the weight of iron, are abstracted for readiness of reference: — 2l8 WEIGHT OF METALS. Wrought Iron, Rolled. I cubic foot, 480 pounds, or 4.29 cwts. I square foot, i inch thick, 40 pounds. I square foot, 3 inches thick, 120 pounds, or 1.07 cwts. . 3 square feet, i inch thick, 120 pounds, or 1.07 cwts. I lineal foot, i inch square, 3 ^ pounds, or .03 cwt I cubic inch, say 0.28 pound. 3.6 cubic inch, i pound. I lineal yard, i inch square, 10 pounds. I lineal foot, 3 inches square, 30 pounds. I lineal foot, 6 inches square, 120 pounds, or 1.07 cwts. I lineal foot, 3 inches by i inch thick, 10 pounds. I lineal foot, ]/i inch in diameter,.... 2 pounds. 1 lineal foot, 2 inches in diameter,... 10.5 pounds. I lineal foot, 6 J^ in. in diameter, about i cwt Cast Iron. I cubic foot, 450 pounds, or 4 cwts. 5 cubic feet, i ton. I square foot, i inch thick, 37.5 pounds. I squarefoot, 3 inches thick (^ cub. ft), 112.5 pounds, or i cwt 3 square feet, i inch thick, 112.5 pounds, or i cwt I cubic inch, 0.26 pound. 3.84 cubic inches, i pound. The Table No. 70 contains the weight of iron and other metals for the following volumes : — I cubic foot I square foot, i inch thick, or ^ji^ih. of a cubic foot I lineal foot, i inch square, or 7xath of a square foot I cubic inch, or Viath of a lineal foot. A sphere, i foot in diameter. The specific gravity due to the respective weights per cubic foot is also given, and likewise the specific weight or heaviness, taking the weight of wrought iron as i, or unity. The next Table, No. 71, contains the volumes of iron and other metals for the following weights : — I ton, in cubic feet I cwt, in square feet, i inch thick. I cwt, in lineal feet, i inch square. I pound, in cubic inches. I ton, as a sphere, in feet of diameter. I ton, as a cube, in feet of lineal dimension. The next Table, No. 72, contains the weight of 1 square foot of metals of various thickness, advancing by sixteenths and by twentieths of an inch, up to I inch in thickness. The fourth Table, No. 73, contains the weight of prisms or bars of iron, and other metals, or metals of any other uniform section, for given sectional areas, varying from .1 square inch to 10 square inches of section, advancing by one-tenth of an inch, for i foot and i yard in length. TABLES Ot WEIGHT AND VOLUME OF METALS. 219 This table is useful in calculations of the weights of bars of every form, rails, joists, beams, girders, tubes, or pipes, &c., when the sectional area is given. The table is available for finding the weight of a metal for any sectional area up to 100 square inches, by simply advancing the decimal points one place to the right; or, in round numbers, up to 1000 square inches, by advancing the decimal points two places. For example, to find the weight of wrought iron having a sectional area of 17 square inches:— For 1.7 square inches, the weight per foot is 5.67 pounds. For 17 square inches, the weight per foot is 56.7 pounds. For 170 square inches, the weight per foot is 567 pounds. Table No. 70. — Weight of Metals. Mbtau Wrought Iron. Cast Iron 1 Steel j Copper, Sheet Copper, Hammered Tin Zinc Lead. Brass, Cast. Brass, Wire, Gon Metal Silver Gold Platinum Cubic Foot. lbs. or cwts. 480 or 450 or '489.6 or 549 or 556 or 462 or 437 or 712 or 505 or 533 or 524 or 655 or 1 200 or 1342 or 4.29 4.02 4-37 4.90 4.96 4.13 3.90 6.36 4.51 4.76 4.68 5.85 10.72 12.00 Square Foot, X inch Thick. lbs. or cwts. 40 or 37.5 or 40. 8 or 45.8 or 46.3 or 38.5 or 36.4 or 59. 3 or 42. 1 or 44.4 or 43- 7 or 54. 6 or loo.oor iii.8or 357 335 364 409 413 344 325 530 375 396 390 488 893 1. 000 Lineal Foot, I Inch Square. lbs. 3.333 3.125 3.400 3-813 3.861 3.208 3.035 4.944 3.507 3.701 3.639 4.549 8.333 9.320 Cubic Inch. lb. .278 .260 .283 .318 .322 .268 •253 .412 .292 .308 ■304 .379 .694 .777 Sphere, I Foot Dia- meter.' lbs. 251 236 257 287 291 242 229 373 264 279 274 343 628 703 Specific Gravity. Water = x. 7.698 7.217 8.805 8.917 7.409 7.008 II. 418 8.099 8.548 8.404 10. 505 19.245 21.522 Specific eight. apei Wei Wro'ght Iron=z. 1.000 .9375 1.020 1. 144 1. 158 .962 .910 1.483 1.052 I.IIO 1.092 1.365 2.500 2.796 Table No. 71. — ^Volume of Metals for given Weights. Metal. Wromgfht Iron Cast &on Steel Copper, Sheet .... Copper, Hammered Tm Zinc Lead Brass, Cast.. Brass, Wire. Gun Metal Silver Gold Platinum Cubic Feet to a Ton. cubic feet 4.67 4.98 4.58 4.08 4.03 4.86 5.13 3.15 4.44 4.20 4.28 3.42 1.87 1.67 Square Feet, X Inch Thick, toacwt. square feet. 2.80 2.99 2.75 2.44 2.42 2.91 3.08 1.89 2.67 2.30 2.56 2.05 1. 12 I.OO Lineal Feet, I In. Square, to a cwt. feet. 33.6 35.8 32.9 29.4 29.0 34.9 36.8 22.7 31.9 30.3 30.8 24.6 13.4 12.0 Cubiclnches to a lb. cubic inches. 3.60 3.84 3.53 3.15 3." 3-74 3-95 2.43 3.42 3.24 3-30 2.64 1.44 1.29 Diameter of a Sphere of I Ton. feet. 2.07 2.12 2.26 1.98 1.98 2.10 2.14 1. 81 2.04 2.00 2.02 1.87 1.59 1.47 Side of a Cube of z Ton. feet. 1.67 I.71 1.66 1.60 1-59 1.69 1.73 1.47 1.64 1. 61 1.62 1.51 1.28 1.19 220 WEIGHT OF METALS. Table No. 72. — Weight of i Square Foot of Metals. Thickness advancing by Sixteenths of an Inch. Thick- ness. inch. 3/16 % s/16 H 7/16 9/16 H "As H '3/16 H 15/16 I Wro't Ikon. Specific wt. =1, lbs. 2.50 5.00 7.50 10. 12.5 150 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 Cast Iron. Specific wt.=.937S. lbs. 2.34 4.69 7.0^ 9.38 II.7 14. 1 16.4 18.7 21. 1 23.5 25.8 28.1 30.5 32.8 35-2 37.5 Stbel. Specific Wt.= 1.02. lbs. 2.55 5.10 7.65 10.2 12.8 »5-3 17.9 20.4 23.0 28.1 30.6 33.2 35-7 38.3 40.8 Copper. Specific wt.=i,i6. lbs. 2.89 5-79 8.68 II. 6 14.5 17.4 20.3 23.2 26.0 28.9 31.8 34.7 37.6 40.5 43.4 46.3 Tin. Specific wt=.962. lbs. 2.41 4.81 7.22 963 12.0 14.4 16.8 19.3 21.7 24.1 26.5 28.9 313 33-7 36.1 38.5 Zinc. Specific wt.=.9io. lbs. 2.28 4.55 6.83 9.10 II.4 137 15.9 18.2 20.5 22.8 25.0 27.3 29.6 31-9 34.1 36.4 Brass. Specific wt. =1.052. lbs. 2.63 5.26 7.89 10.5 13.2 il8 18.4 21. 1 237 26.3 28.9 31.6 34.2 36.8 39-5 42.1 Gun Metal. Specific wt.= 1.092. lbs. 2.73 5.46 8.19 10.9 13-7 16.4 19. 1 21.9 24.6 27.3 30.0 32.8 35.0 38.2 41.0 43-7 Lead. Specific wt.=i.48. lbs. 3.71 7.41 II. I 14.8 18.5 22.2 25.9 29.7 33-4 37. i 40.8 44.5 48.2 519 55-6 59-3 Thickness advancing by Twentieths of an Inch. inch. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. .05 2.00 1.88 2.04 2.32 '•§3 1.82 2. II 2.19 2.96 .10 4.00 3.75 4.08 4.63 3.85 3-64 4.21 4.37 5.93 .15 6.00 5.63 6.12 6.95 5.78 5.46 6.32 6.56 8.90 .20 8.00 7.50 3.16 9.26 7.70 7.28 8.42 8.74 "•§ .25 10.0 9.38 10.2 II. 6 9.63 9.10 10.5 10.9 14^ .30 12.0 "3 12.2 13.9 II. 6 10.9 12.6 131 17.8 .35 14.0 131 14.3 16.2 135 12.7 14-7 15.3 20.8 .40 16.0 15.0 16.3 18.5 154 14.6 16.8 17.5 23.7 .45 18.0 16.9 18.4 20.8 17.3 16.4 18.9 19.7 26.7 .50 20.0 18.8 20.4 23.2 19-3 18.2 21. 1 21.9 29.7 .55 22.0 20.6 22.4 25-S 21.2 20.0 23.2 24.0 32-7 .60 24.0 22.5 245 27.8 23.1 21.8 25.3 26.2 35.6 .65 26.0 24.4 26.5 30.1 25.0 23-7 27.4 28.4 38.6 .70 28.0 26.3 28.6 32.4 27.0 255 29.5 30.6 41.5 •75 30.0 28.1 30.6 34.7 28.9 27.3 31.6 32.8 44-5 .80 32.0 30.0 32.6 37.0 30.8 29.1 33-7 35.0 47.5 .85 34.0 31-9 34.7 39.4 32.7 30.9 35.8 37.2 50.4 .90 36.0 33.8 ^tl 41.7 34.7 32.8 37-9 39.3 53-4 .95 38.0 35-6 38.8 44.0 36.6 34-6 40.0 41.5 56.3 I.OO 40.0 37.5 40.8 46.3 38.5 36.4 42.1 43-7 59.3 Note to Table *j% next page. — To find the weight of I lineal foot or I lineal yard of hammered iron, copper, tin, zinc, or lead, multiply the tabular weight for rolled wrought iron of the given dimensions by the following multipliers, respectively : — Exact. Approximate. Hammered Iron 1.008 i.oi equivalent to I percent more. Copper 1. 158 1. 16 ,, 16 ,, more. Tin 962 96 ,, 4 ,, less. Zinc 91 91 ,, 9 „ less. Lead I'483 1.48 ,, 48 „ more. WEIGHT OF METALS OF A GIVEN SECTIONAL AREA. 221 Table No. 73. — Weight of Metals, of a given Sectional Area, PER Lineal Foot and per Lineal Yard. Rolled Wrought Iron. Cast Iron. Stesl. Brass. Gun Mbtal. Sect. ' Akea Sp. Wcight=x. Sp.Weight=.9375. Sp. Weight=i.o2. Sp.Weight-=i.o52. Sp» Weight= 1.092. 1 I Foot. I Yard. X Foot. I Yard. I Foot. I Yard. I Foot 1 Yard. I Foot.* I Yard. , sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. .1 -333 I.OO •313 .938 .340 1.02 •351 1.05 ■364 1.09 .2 .667 2.00 .625 1.88 .680 2.04 .701 2.10 .728 2.18 •3 I.OO 3.00 .935 2.81 1.02 3.06 1.05 3.16 1.09 3.28 .4 1-33 4.00 1.25 3.75 1.36 4.08 1.43 4.21 1.46 4-37 •5 1.67 5.00 1.56 4.69 1.70 5- 10 1.75 5.26 1.82 5.46 .6 2.00 6.00 1.88 5.63 2.04 6.12 2.II 6.31 2.18 6.55 .7 2-33 7.00 2.19 6.56 2.38 7.14 2.46 736 2.55 7.64 .8 2.67 8.00 2.50 7.50 2.72 8.16 2.81 8.42 2.91 8.74 •9 3.00 9.00 2.81 8.44 3.06 9.18 3.16 9-47 3.28 9.83 1 1.0 3.33 10. 3.15 9.38 3.40 10.2 3.51 I0.5 364 10.9 I.I 3-67 II.O 3-44 10.3 3-74 II. 2 3.86 II. 6 4.00 12.0 1.2 4.00 12.0 3-75 "3 4.08 12.2 4.21 12.6 4-37 13.1 1-3 4.33 130 4.06 12.2 4.42 13-3 4,56 13-7 4.73 14.2 1.4 4.67 14.0 4.38 13- 1. 4.76 14.3 4.91 14.7 5.10 15.3 i-S 5.00 15.0 4.69 14. 1 S'lo 15-3 5.26 15.8 5.46 16.4 1.6 5.33 5.67 16.0 5.00 15.0 5-44 16.3 5.61 16.8 5.82 >7-5 1-7 17.0 5.31 159 5.78 17.3 5.96 17.9 6.19 18.6 1.8 6.00 18.0 563 16.9 6.12 18.4 6.31 18.9 6.55 19-7 1-9 6.33 19.0 5-94 17.8 6.46 19.4 6.66 20.0 6.92 20.8 2.0 6.67 20.0 6.25 18.8 6.80 20.4 7.01 21.0 7.28 21.8 2.1 7.00 21.0 6.56 19.7 7.14 21.4 736 22.1 7.64 22.9 2.2 7.33 22.0 6.88 20.6 7.48 22.4 7.72 23.1 8.01 24.0 2-3 7.67 23.0 7.19 21.6 7.82 235 8.07 24.2 8.37 25.1 2.4 8.00 24.0 7.50 22.5 8.16 24.5 8.42 253 8.74 26.2 2.5 f.33 25.0 7.81 23-4 8.50 255 8.77 26.3 9.10 27.3 2.6 8-67 26.0 8.13 24.4 8.84 26.5 9.12 27.4 9.46 28.4 H 9.00 27.0 8.44 253 9.18 27.5 9.47 28.4 9.83 29.5 2.8 9.33 28.0 8.75 26.3 9.52 28.6 9.82 29.5 10.2 30.6 2.9 9.67 29.0 9.06 27.2 9.86 29.6 10.2 30s 31.6 10.6 ^H 3.0 lO.O 30.0 9.38 28.1 10.2 30.6 10.5 10.9 32.8 31 10.3 310 9.69 29.1 10.5 31.6 10.9 32.6 "3 33-9 3-2 10.7 32.0 10.0 30.0 10.9 32.6 II. 2 33-7 11.7 34.9 3-3 II.O 330 10.3 309 II. 2 33-7 I1.6 34.7 12.0 36.0 3-4 "•3 34.0 10.6 31-9 11.6 34.7 11.9 35.8 12.4 37.1 3.5 1 1.7 35.0 10.9 32.8 11.9 35-7 12.3 36.8 12.7 38.2 3.6 12.0 36.0 "3 33.8 12.2 36.7 12.6 37.9 13.1 39.3 3.7 *2.3 37.0 11.6 34.7 12.6 37.7 130 38.9 13.5 40.4 3.8 12,7 380 11.9 35.6 12.9 38.8 133 40.0 13.8 41.5 3.9 13.0 39.0 12.2 36.6 133 39-8 13.7 41.0 14.2 42.6 4.0 »3.3 40.0 12.5 37.S 13.6 40.8 14.0 42.1 14.6 43.7 4.1 13.7 41.0 12.8 38.4 139 41.8 14.4 43-1 14.9 44.8 4-2 14.0 42.0 131 39-4 14.3 42.8 14.7 44.2 15.3 45.9 4.3 H3 43.0 ^H 40.3 14.6 43.9 15. 1 45.2 15.7 46.9 4-4 14.7 44.0 13-8 41.3 15.0 44.9 ^H 46.3 16.0 48.0 4-5 15.0 45.0 14. 1 42.2 15.3 45-9 15.8 4Z-3 16.4 49.1 4.6 15.3 46.0 14.4 43-1 15.6 46.9 16. 1 48.4 16.7 50.2 4.7 15.7 47.0 14.7 44.1 16.0 47.9 16.5 49.4 17. 1 51.3 4.8 16.0 48.0 15.0 45.0 16.3 49.0 16.8 50.5 17.5 52.4 4.9 16.3 49.0 15-3 45.9 16.7 50.0 17.2 51.6 17.8 53.5 5.0 16.7 50.0 15.6 46.9 17.0 51.0 17.5 52.6 18.2 54.6 222 WEIGHT OF METALS. Table No 73 {continued). R0M.RD Wrought Irom. Cast Iron. Stbel. Brass. Gun Metal. Sect. Sp. Weight=x. Sp.Weight — 9375- Sp.Weight=z.03. Sp.Weight=x.o53. Sp.Weight=x.o9a. Akra mmW^M»*\» xFoot xYard. I Foot xYard. I Foot xYard. I Foot xVard. X Foot I Yard. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 5.1 17.0 51.0 15.9 47.8 17.3 52.0 17.9 53-7 18.6 m 5.2 17.3 52.0 16.3 48.8 17.7 53- 18.2 54- Z 18.9 5-3 17.7 53.0 16.6 49.7 18.0 54.1 18.6 55.8 X9-3 57-9 5.4 18.0 54.0 16.9 50.6 18.4 55.1 18.9 56.8 19.7 58.9 55 18.3 55.0 17.2 51.6 18.7 56.1 19.3 57.9 58.9 20.0 60.0 5.6 18.7 56.0 17.5 .52.5 19.0 57-^ 19.6 20.4 20.8 61. 1 5.7 19.0 57.0 17.8 53.4 19.4 58.1 20.0 60.0 62.2 5.8 19.3 58.0 18. 1 54-4 19.7 59.2 20.3 61.0 21.1 633 5-9 19.7 59.0 18.4 55.3 20.1 60.2 20.7 62.1 21.5 21.8 64.4 6.0 20.0 60.0 18.8 56.3 20.4 61.2 21.0 63.1 65.5 6.1 20.3 61.0 19. 1 57.2 20.7 62.2 21.4 64.2 22.2 66.6 6.2 20.7 62.0 19.4 58.1 21. 1 63.2 21.7 65.2 22.6 67.7 6.3 21.0 63.0 197 59.1 21.4 64.3 22.1 66.3 22.9 68.8 6.4 21.3 64.0 20.0 60.0 21.8 65.3 22.4 67.3 23.3 69.9 6.5 21.7 65.0 20.3 60.9 22.1 66.3 22.8 68.4 23.7 70.9 6.6 22.0 66.0 20.6 61.9 22.4 67.3 23.1 69.4 24.0 72.0 H 22.3 67.0 20.9 62.8 22.8 68.3 23.5 70.5 24.4 24.8 731 6.8 22.7 68.0 21.3 63.8 23.1 69.4 23.9 71.5 74.2 6.9 23.0 69.0 21.6 64.7 23.5 70.4 24.2 72.6 25.1 75-3 7.0 23.3 70.0 21.9 65.6 23.8 71.4 24.6 73.6 25.5 76.4 7.1 23.7 71.0 22.2 66.6 24.1 72.4 24.9 74.7 25.8 77-5 7.2 24.0 72.0 22.5 67.5 24,5 73.4 25.3 75-7 26.2 78.6 7.3 24.3 730 22.8 68.4 24.8 74.5 25.6 76.8 26.6 IH 7.4 24.7 74.0 23.1 69.4 25.2 75.5 26.0 77.9 26.9 80.8 7.5 25.0 75.0 ^H 70.3 25.5 76.5 26.3 78.9 27.3 81.9 7.6 253 76.0 23.8 71.3 25.9 77-5 26.7 80.0 27.7 83.0 7.7 25.7 77.0 24.1 72.2 26.2 78.5 27.0 81.0 28.0 84.1 7.8 26.0 78.0 24.4 73.1 26.5 79.6 27.4 82.1 ^?i 85.2 7.9 26.3 79.0 24.7 74.1 26.9 80.6 27.7 83.1 28.8 86.3 8.0 26.7 80.0 25.0 75.0 27.2 81.6 28.1 84.2 29.1 87.4 8.1 27.0 81.0 253 75-9 27.5 82.6 28.4 85.2 295 88.5 8.2 27.3 82.0 25.6 76.9 27.9 83.6 28.8 86.3 29.9 89.5 8.3 27.7 83.0 25.9 77.8 28.2 84.7 29.1 111 302 90.6 8.4 28.0 84.0 26.3 78.8 28.6 85.7 29.5 29.8 30-6 91.7 !-5 28.3 85.0 26.6 79.7 28.9 86.7 89.4 30.9 92.8 8.6 28.7 86.0 26.9 80.6 29.2 ll^ 30.2 90.S 31-3 93-9 8.8 29.0 5Z-° 27.2 81.6 29.6 IH 30.5 91.5 31.7 95.0 29.3 88.0 27.5 82.5 29.9 89.8 30.9 92.6 32.0 96.1 8.9 29.7 89.0 27.8 83.4 30.3 90.8 31.2 93-6 32.4 97.2 9.0 30.0 90.0 28.1 84.4 30.6 91.8 316 94.7 32.8 98.3 9.1 30.3 91.0 28.4 !l-3 309 92.8 31-9 957 33.x 99.4 9.2 30.7 92.0 28.8 86.3 31-3 93.8 32.3 96.8 33.5 100.5 9.3 31.0 930 29.1 87.2 31.6 94.9 32.6 97.8 33.9 101.6 9.4 31.3 94.0 29.4 88. z 32.0 95-9 33.0 98.9 34-2 102.7 ^•5 31.7 95.0 29.7 89.1 32.3 96.9 33-3 99.9 34-6 X03.7 9.6 32.0 96.0 30.0 90.0 32.6 97.9 33.7 lOI.O 34.9 104. S H 32.3 97.0 30.3 90.9 33'0 98.9 34-0 102.0 35.3 105.9 9.8 32.7 98.0 30.6 91.9 33.3 100. 34.4 103. 1 35.7 107.0 9.9 33.0 99.0 30.9 92.8 33.7 lOI.O 34.7 104.2 36.0 108. 1 lO.O 33-3 100. 31.3 93.8 34-0 102.0 35.1 105.2 36.4 109.2 See note at foot of page 220. RULES FOR WEIGHT. 223 Rules for the Weight of Iron and Steel. The following rules for finding the weight of wrought iron, cast iron, and steel, are based on the data contained in Tables No. 70 and 71. Rule i. — ^To find the Weight of Iron or Steel, when the volume in cubic feet is given. Multiply the volume by 4.29 for wrought iron, 4.02 for cast iron, 4.37 for steel. The product is the w«ight in hundredweights. Rule 2. — When the volume in cubic inches is giveny multiply the volume by .278 (or .28) for wrought iron, .26 for cast iron, .283 for steel. The product is the weight in pounds. Rule 3. — WAen the quantity is reduced to square feet, one inch in thickness^ multiply the area by 40 for wrought iron, 37/^ for cast iron, 40.8 (or 41) for steel. The product is the weight in pounds. Or, multiply the area by •357 ^or wrought iron, •335 ^or cast iron, .364 for steel. The product is the weight in hundredweights. Rule 4. — When the sectional area in square inches, and t/ie length in feet^ of a bar or prism are givcfi, multiply the sectional area by the length, and by 3 V3 for wrought iron, 3^ for cast iron, 3.4 for steel. The product is the weight in pounds. For large masses, multiply the sectional area by the length, and divide the product by 672 for wrought iron, 717 for cast iron, 659 for steel. The quotient is the weight in tons. Rule 5. — W/ien the sectional area in square inches , and the length in yards , of a barorprisniy are given, multiply the sectional area by the length, and by 10 for wrought iron, 9.375 for cast iron, 10.2 for steel. The product is the weight in pounds. 224 WEIGHT OF METALS. Rule 6. — To find the sectional area of a bar or prism of iron OR steel, when the length and tfu toted weight are given. Divide the weight in pounds by the length in feet, and by 3 ^3 for wrought iron, 3^ for cast iron, 3.4 for steel. The quotient is the sectional area in square inches. Rule 7. — To find the length of a bar, prism, or other piece of uniform section of iron or steel, whefi the total weight and the sectional area are given. Divide the weight in pounds by the sectional area in square inches, and by 3 '/3 for wrought iron, 3 ^ for cast iron, 3.4 for steel. The quotient is the length in feet. In applying the last rule to calculate the length of wire of a given size, for a given weight, say i cwt. of wire, the sectional area of the wire is found, in the usual way, by multiplying the square of the thickness or diameter, //, by .7854. Then, by the rule, the length in feet of i cwt. of iron wire is equal to 112 42.78 In the same way, the dividends of the fractions to express the length of I cwt. of other metals may be found, and the following is a special rule lor wire : — Rule 8. — To find the length of one hundredweight of wire OF A given thickness. Divide the following numbers by the square of the diameter or thickness, in parts of an inch : — 42.78 for wrought iron, 42 for steel, 37.43 for copper, 38.54 for brass, 31.34 for silver, 17.12 for gold, 15.28 for platinum. The quotient is the length in feet. Note. — This rule may be used for finding the weight of round bar iron. 2. It is known that the density of wire is not perfectly constant, but that there is some degree of variation, according to the size. It is generally- understood that the density is reduced as the wire is drawn smaller, but it appears from the table of the weight of Warrington wire, that the density is greater as the size is less. The same inference is to be drawn from tabular statements of the length of one kilogramme of wire according to the French gauge (Table No. 31, page 148). One of these statements is given on the next page, from which it is apparent that the length of iron RULES FOR WEIGHT. 225 required to weigh a kilogramme decreases more rapidly than the sectional area increases. For example, the diameter being 6, 12, 24, 30 tenths of a millimetre, the squares of which, or the relative volumes of a given length, are as I. 4, 16, 25; the lengths of a kilogramme are 405» ii5> 3o> 20 metres, which are inversely as ^ 3-5» i3-5> 20.2. Showing that a shorter length is required in proportion to the volume, as the diameter of the wire is reduced, and that the density of the smaller wire must therefore be the greater. Table No. 73^. — ^Weight of Galvanized Iron Wire (French). Ka. of Gauge. Diameter. Length of X Kilogramme. | No. of Gauge. Diameter. Length of z Kilogramme. millimetres. metres. millimetres. metres. I 0.6 405 13 0.20 40 2 0.7 370 14 0.22 35 3 0.8 260 15 0.24 30 4 0.9 215 16 0.27 25 5 O.IO 175 17 0.30 20 • 6 O.I I 140 18 0.34 15 7 0.12 115 19 0-39 10 8 0.13 103 20 0.44 9 9 0.14 82 21 0.49 6 10 0-15 70 22 0.54 5 II 0.16 65 23 0-59 4 12 0.18 50 3. The densities of metals assumed in the foregoing rules are those which are tabulated in Table No. 65. 4. In estimating the weight of cast iron from plans, the weight is fre- quently calculated at the same rate as for wrought iron, which is heavier ^n cast iron, with the object of providing an allowance, by way of com- pensation, for occasional swellings or enlargements of castings in excess of '^''•e exact dimensions of patterns. The following tables of the weight of metals in various forms have been 'Collated by means of the preceding rules. The sectional areas of bars ~id other pieces of uniform section are, in some tables, added for each ^^ntling. The length of bar, and the area of plates and sheets, required '"^ weigh I cwt., or i ton, are given. 10 226 WEIGHT OF METALS. LIST OF TABLES OF THE WEIGHT OF WROUGHT IRON, In Bars, Plates, Sheets, Hoop-iron, Wire, and Tubes. Table No. 74. — Weight of Flat Bar Iron; width, i to 11 inches; thick- ness, 7x6 to I inch; length, i to 9 feet. Table No. 75. — Weight of Square Iron; ^ to 6 inches square; length, I to 9 feet Table No. 76. — ^Weight of Round Iron, ^ to 24 inches in diameter; length, I to 9 feet Table No. 77.— Weight of Angle-Iron and Tee-Iron; sum of the width and depth, ly^ to 20 inches; thickness, J^ to i inch; length, i foot In the composition of this table, it has been assumed that the base and the web or flange are of equal thicknesses; and that the reduction of area of section by rounding off the edges, is compensated by the filling in at the root of the flange. Table No. 78. — ^Weight of Wrought-iron Plates; area, i to 9 square feet; thickness, X to 15 inches. Table No. 79. — Weight of Sheet Iron, according to wire-gauge used by South Staffordshire sheet-rollers; area, i to 9 square feet; thickness. No. i to No. 32 wire-gauge. Table No. 80. — Weight of Black and Galvanized Iron Sheets (Morton's Table). Table No. 81. — Weight of Hoop Iron; width, ^ to 3 inches; thickness. No. 4 to No. 21 wire-gauge; length, i foot Table No. 82. — ^Weight and Strength of Warrington Iron Wire. Table No. 83. — Weight of Wrought-iron Tubes, by internal diameter; diameter, f^ to 36 inches; thickness, ^ inch to No. 18 wire-gauge; length, I foot . Table No. 84. — Weight of Wrought-iron Tubes, by external diameter; diameter, i to 10 inches; thickness, No. 15 wire-gauge to s/x6 inch; length, I foot Multipliers, derived from table No. 70, are subjoined, by which the tabulated weights of wrought iron may be multiplied, in order to find from these tables the weight of bars, plates, or sheets of other metal. — Multipliers. Hammered Iron i.oi Cast Iron 94 Steel ; 1.02 Sheet Copper 1.14 Hammered Copper 1.16 Lead 1.48 Cast Brass 1.05 Brass Wire i.ii Gun Metal 1.09 FLAT BAR IRON. 227 Table No. 74— WEIGHT OF FLAT BAR IRON. I INCH Wide. Thick- xass. Sbct. AXSA. Length in ] Feet. Len^h to weigh X cwt. z 2 3 4 5 6- 7 8 9 iacbes. sq. in. lbs. lbs. lbs. lbs. lbs. ll». lbs. lbs. lbs. feet H 5/16 H llt6 -250 .313 .438 .500 .833 1.04 1.25 1.46 1.67 1.67 2.08 2.50 2.92 3-33 2.50 3.12 3-75 4.38 5.00 3.33 4.16 5.00 5.84 6.67 4.17 5.20 6.25 7.29 8.33 5.00 6.24 7.50 8.76 lO.O 5.83 7.28 8.75 10.2 II.7 6.67 8.32 10. II.7 13-3 7.50 9.36 "3 13. 1 15.0 134.4 89.6 76.8 67.2 9/t6 "/i6 •5^3 .750 1.88 2.08 2.29 2.50 3.75 4.16 4.58 5.00 5.62 6.25 6.87 7.50 7.50 8.33 9.17 lO.O 9.37 10.4 II.4 12.5 "•3 12.5 13.8 15.0 13- 1 14.6 16.0 17.5 15.0 16.6 18.3 20.0 16.9 18.8 20.6 22.5 597 53-8 48.9 44.8 I .813 .875 -938 1. 00 2.71 2.92 3.13 3.33 5.42 5.84 6.25 6.67 8.12 8.76 9.38 10.0 10.8 II. 7 12.5 13.3 13.5 14.6 15.6 16.7 16.3 18.8 20.0 19.0 20.4 21.9 23.3 21.7 23.4 25.0 26.7 24.4 26.3 28.1 30.0 41.4 38.4 35.8 33.6 i}i INCHES Wide. 1 iacbes. sq. in. V .2S1 S/jfi .352 H .422 7/16 .492 H .563 »/t« .633 H .703 r .844 r .914 .984 '5/16 1.06 I I.I3 lbs. .938 I.17 I.4I 1.64 1.88 2. II 2.34 2.58 2.91 3.05 3.28 3.52 3-75 lbs. 1.88 2.34 2.S1 3.28 3.75 422 4.69 5.16 5.63 6.09 6.56 7.03 7.50 lbs. 2.81 3.52 4.22 4.92 5.62 6.33 7.03 7.73 8.44 9.14 9.84 10.6 "•3 lbs. 3.75 4.68 5.62 6.56 7.50 8.44 9.38 10.3 "3 12.2 13. 1 14. 1 15-0 lbs. 4.69 5.86 7.03 8.20 9.38 10.6 n.7 12.9 14.0 15.2 16.4 17.6 18.8 lbs. 5.63 7.03 8.44 9.84 "3 12.7 13.1 15.5 16.9 18.3 19.7 21. 1 22.5 lbs. 6.56 8.20 9.84 II.5 13.1 14.8 16.4 18.0 19.7 21.3 23.0 24.6 26.3 lbs. 7.50 9.37 "•3 13. 1 15.0 16.9 18.8 20.6 22.5 24.4 26.3 28.1 30.0 lbs. 8.44 10.6 12.7 14.8 16.9 19.0 21. 1 23.2 25.3 27.4 29.5 31.6 33.8 1% INCHES Wide. lbs. 3.12 3.91 4.69 5.47 6.25 7.03 7.81 8.59 9.38 10.2 10.9 II. 7 12.5 lbs. 4.17 5.21 6.25 7.29 8.33 9.38 10.4 II.5 12.5 13.5 14.6 15.6 16.7 lbs. 5.21 6.51 7.81 9.12 10.4 II.7 13.0 14.3 15.6 16.9 18.2 20.8 lbs. 6.25 7.82 9.38 10.9 12.5 14. 1 15.6 17.2 18.8 20.3 21.9 23.4 25.0 lbs. 7.29 9. 1 1 10.9 12.8 14.6 16.4 18.2 20.1 21.9 237 25-5 27.3 29.2 lbs. 8.33 10.4 12.5 14.6 16.7 18.8 20.8 22.9 25.0 27.1 29.2 31.2 33.3 lbs. 9.37 II. 7 14. 1 16.4 18.8 14. 1 23.4 2J.8 28.1 30.5 32.8 35.1 37.5 feel. II9.5 95.6 79.6 68.3 59.7 53-1 47.8 43-4 39-8 36.8 34.1 31.9 29.9 feet. 107.5 94.0 71.7 61.2 53.8 47.8 43.0 39.1 35.8 33.' 30.7 28.7 26.9 228 WEIGHT OF METALS. Weight of Flat Bar Iron. i)i INCHES Wide. Length in Feet. Thick- ness. Sect. Area. Len^ to weigh z a 3 4 5 6 7 8 9 X CWL inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X .344 1.15 2.29 3.44 4.58 5.73 6.87 8.02 9.17 10.3 97.7 S/i6 .430 1-43 2.86 4.30 5.73 7.16 8.59 10. II.5 12.9 78.2 H .516 1.72 3-44 5.16 6.87 8.59 10.3 12.0 13-7 '$■5 65.6 7/16 .602 2.01 4.01 6.02 8.02 10. 12.0 14.0 16.0 18.0 48."9 }i .688 2.29 4.58 6.87 9.17 "•5 13-8 16.0 18.3 2a 6 9/16 •273 .859 2.58 2.86 5.16 7.73 10.3 12.9 15.5 18.0 20.6 1 23.2 43-4 H 5.73 8.59 II.5 '4-3 17.2 20.1 22.9 25.8 39.1 r .945 315 6.31 6.88 9-45 12.6 15.8 18.9 22.1 25.2 28.4 35-5 1.03 3.44 10.3 13.8 17.2 20.6 24.1 27.5 309 •32.6 »3/i6 1. 12 3-72 8.02 II. 2 14.9 18.6 22.3 26.1 29.8 33.5 30.1 'A 1.20 4.01 12.0 16.0 20.0 24.1 28.1 32.1 36.1 27.9 »s/i6 1.29 4.30 8.59 12.9 17.2 21.5 25.8 30.1 34-4 38.7 26.1 I i.3» 4.58 9.17 13.8 18.3 22.9 27.5 32.1 36.7 41.3 24.4 i}i INCHES Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X .275 1.25 •2.50 3-75 5.00 6.25 7.50 8.75 10. "•3 89.6 s/16 .469 1.56 3.13 4.69 6.25 7.82 9.38 10.9 12-5 14. 1 78.3 H .563 1.88 3-75 5.63 7.50 8.75 9.38 IL3 13. 1 150 16.9 59.7 7/16 .656 2.19 4.38 6.56 10.9 13. 1 15.3 17.5 19.7 5*-; H .750 2.50 5.00 7.50 10.0 12.5 15.0 17.5 20.0 22.5 44.8 9/16 .844 2.81 5.63 8.44 "•3 14. 1 16.0 19.7 22.5 25.3 39.8 ^, .938 3.13 6.25 6.88 9.38 12. J 15.6 18.8 21.9 25.0 28.1 35.8 r 1.03 3-44 10.3 13.8 17.2 20.6 24.1 27.5 30.9 32.6 i»3 3.75 7.50 "3 15.0 18.8 22.5 26.3 30-0 33.8 29.9 «3/i6 1.22 4.06 8.13 12.2 16.3 20.3 24.4 28.4 32.5 36.6 27.6 ^. I-3I 4.38 8.75 131 17.J 21.9 26.3 30.6 350 39.4 25.6 »s/i6 1.41 4.69 9-38 14. 1 18.8 23.4 28.1 32.8 37.5 42.2 23.9 I 1.50 5.00 lO.O 15.0 20.0 25.0 30.0 350 40.0 45.0 22.4 i^ INCHES Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X .406 1-35 2.71 4.06 5-41 6.8 8.10 9.48 10.8' 12.2 82.7 s/16 .508 1.69 3-39 5.07 6.77 8.5 10.2 II.8 13.5 15.2 66.2 H .boq 2.03 4.06 6.09 8.12 10.2 12.2 14.2 16.2 18-3 55.1 7/16 .813 2.37 4-74 7.II 8.12 9.48 11.8 14.2 16.6 19.0 21.3 47.3 ^ 2.71 5.42 10.8 13-5 16.2 19.0 21.6 24-4 41.3 9/16 .914 3.05 6.09 9.14 12.2 ^i-^ 18.3 21.3 24.4 27.4 36.8 ^, 1.02 3-39 6.77 10.2 13.5 16.9 20.3 23.7 ^H 30.5 33.1 "A6 1. 12 3-72 7.45 II. 2 14.9 18.6 22.3 26.1 29.8 33-5 30.1 H 1.22 4.06 8.13 12.2 16.3 20.3 24.4 28.4 32.5 36.6 27.6 »3/i6 1.32 4.40 8.80 132 17.6 22.0 26.4 30.8 35-2 39.6 25.4 ^. L43 4.74 9.48 14.2 19.0 23.7 28.4 33.2 37.9 42.7 23.6 ^s/t6 I.S3 5.08 10.2 15.2 20.3 254 30.5 35.5 40.6 48.8 22.1 I 1.63 5.42 10.8 16.3 21.7 27.1 32.5 37-9 43.3 21.2 FLAT BAR IRON. 229 Weight of Flat Bar Iron. 1^ INCHES Wide. Length in Fbbt. Thioc- NESS. Sbct. Aksa. Length to weigh Z 2 3 4 5 6 7 8 9 X cwt. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet X .638 1.46 2.92 4.37 5.83 7.29 8.74 10.2 II.7 13. 1 76.8 5/16 .547 1.82 3.6J §-^? 7.29 9.II 10.9 12.8 14.6 16.4 61.4 H .656 2.19 4.38 6.56 8.75 10.9 13.1 15-3 ns 19.7 51.2 7/16 .766 2.55 H^ 7.66 10.2 12.8 15.3 17.9 20.4 23.0 43.9 a .875 2.92 5.«3 8.75 II.7 14.6 17.5 20.4 23.3 26.2 38.4 9/16 .984 3.28 6.56 9.84 13. 1 16.4 19.7 23.0 26.2 29.5 34.1 H 1.09 3.65 7.29 10.9 14.6 19.2 21.9 2$-5 29.2 32.8 •30-7 r 1.20 4.01 8.02 12.0 16.0 20.0 24.1 28.1 32.1 36.1 27.9 I.3I 4.3« 8.75 13.1 17.5 21.9 26.3 306 350 39.4 25.6 r 1.42 1.53 4.74 5.10 9.48 10.2 14.2 15.3 19.0 20.4 23.7 25. S 28.4 30.6 33.2 35-7 37.9 40.8 43-2 45.9 23.7 21.9 ^s/i6 1.64 5-47 10.9 16.4 21.9 27.3 32.8 1? ) 1 40.8 43-7 49.2 20.5 I 1.75 5.«3 II. 7 17.5 233 29.2 35.0 46.7 52.5 19.2 iji INCHES Wide. tnf|<^^ 1 sq. in. ' .469 lbs. 1.56 lbs. 3.13 lbs. 4.69 lbs. 6.2$ lbs. 7.81 lbs. 9.38 lbs. 10.9 lbs. 12.5 lbs. 14. 1 feet 71.7 '4 5/.6 1 .586 1.95 3-91 5.86 7.81 9.66 11.7 13.7 15.6 17.6 57-3 > H .703 2.34 4.69 7.03 9.37 II.7 14. 1 16.4 18.8 21. 1 47.8 r/i6 1 .820 2.73 5.47 8.20 10.9 13.7 16.4 19.1 21.9 24.6 41.0 }i 1 .938 313 6.25 9.38 12.5 15-6 18.8 21.9 25.0 28.1 35.8 9/16 1.06 3.52 7.03 10.5 14. 1 17.6 21. 1 24.6 28.1 31.6 31.8 H 1. 17 3-9" 7.81 II.7 14.6 19-5 23.4 27.3 31.2 35.2 28.7 "/•« 1.29 430 8.59 12.9 17.2 21.5 2J.8 28.1 30.1 34.4 38.7 26.1 ' )i I.4I 4.69 9.38 14. 1 18.8 234 32.8 37.5 42.2 239 : »Vi6 ! 1.52 5.08 10.2 15.2 20.3 25.4 30.5 35-5 40.6 45.7 22.1 H 1 1.64 t^ 10.9 16.4 21.9 27.3 32.8 38.3 43.9 49.4 20.5 1 '5/i« 1.76 11.7 17.6 23.4 29.3 35.1 41.0 46.9 52.7 19. 1 1' J.88 6.25 12.5 18.8 25.0 3>.3 37.5 43.8 50.0 56.2 17.9 2 INCHES Wide. indies. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet j( .500 1.67 3-33 5.00 6.67 8.33 10.0 II.7 13.3 15-O 67.2 s/i6 .625 2.08 4.17 6.25 8.33 10.4 12.5 14.6 16.7 18.8 53.8 H 1 .750 2.50 5.00 7.50 lO.O 12.5 15.0 17.5 20.0 22.5 44.8 7/.6 -875 2.92 5.83 8.75 II.7 14.6 17.5 20.4 23.3 26.3 38.4 >^ I.OO 3-33 6.67 10.0 133 16.7 20.0 23.3 26.7 300 33-6 9/i« 1 1. 13 3.75 7.50 "3 15.0 18.8 22.5 26.3 300 33-8 29.9 X, 1 1.38 4.17 8.33 12.5 13.8 16.7 20.8 25.0 29.2 33.3 37.5 26.9 r 4.58 9.16 18.3 22.9 27.5 32.1 36.7 41.2 24.4 1.50 5.00 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 22.4 «3/«6 ' 1.63 5J^ 10.8 16.3 21.7 27.2 32. s 37.9 43.3 48.8 20.7 H m S.83 11.7 '7-5 18.8 23.3 29.2 35.0 40.8 46.7 52.5 19.2 «5/x6 6.25 12.5 25.0 31.3 37.5 43.8 50.0 56.3 17.9 I 2.00 6.67 13.3 20.0 26.7 33.3 40.0 46.7 52.2 60.0 16.8 230 WEIGHT OF METALS. Weight of Flat Bar Iron. 2>i INCHES Wide. Length in Febt. Thick- ness. Sect. Area. Len^h to weigh z 2 3 4 5 6 7 8 9 I cwt. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet V .531 1.77 3.54 I'V 7.08 8.85 10.6 12.4 14.2 15.9 63.2 5/i« .664 2.21 4.43 6.64 8.85 II.7 13.3 ^5-5 17.7 19.9 50.6 H .797 2.66 5-31 7.97 10. D 13.3 '5-9 i8.6 21.2 23.9 42.2 7/16 .930 3.10 6.20 930 12.4 15.5 18.6 21.7 24.8 27.9 36.1 }i 1.06 3.54 7.08 10.6 14.2 17.7 21.3 24.8 28.3 31.9 31.6 9/16 1.20 3.98 7.97 12.0 15.9 20.0 23.9 27.9 31.9 35-5 28.1 ^. 1.33 4.43 8.85 13.3 17.7 22.1 1 26.6 31.0 35.4 39.8 25.3 r 1.46 4.87 9.74 14.6 19.5 24.4 1 29.2 34.1 39- 43.8 23.0 1.59 5.3« 10.6 "59 21.2 26.6 31.9 37.2 42.5 47.8 21. 1 »3/x6 1.74 5.76 11.5 'H 23.0 28.8 34.5 40.3 46.0 51.8 19.8 H 1.86 6.20 12.4 18.6 24.8 31.0 37-; 43.4 49.6 55.8 18. 1 »5/i6 1.98 6.64 13.3 19.9 26.6 33.2 39.8 46.5 53.1 59.8 16.9 I 2.13 7.08 14.2 21.3 28.3 35.4 42.5 49.6 56.7 63.8 15.8 2)4 INCHES Wide. 2f^ inches Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X .563 1.88 3.75 5.63 7.50 9.4 "3 13-1 15.0 16.9 59.7 5/16 .703 2.34 4.69 7.03 9.38 II.7 14. 1 16.4 18.8 21.1 47.8 ^ .844 2.81 5.63 8.44 "3 14. 1 16.9 19.7 22.5 253 39.8 7/16 .984 3.28 6.56 9.84 13. 1 16.4 19.7 23.0 26.3 29.5 33.8 34.1 ^ I.I3 3.75 7.50 11.3 15.0 18.8 22.5 26.3 30.0 29.9 9/16 1.27 4.22 8.44 12.7 16.9 21. 1 ^5-3 29.5 33.8 38.0 26.5 ^, I.4I 4.69 9.38 14. 1 18.8 23.4 28.1 32.8 37.5 42.2 23.9 r 1.55 5.16 10.3 15.5 20.6 25.8 30.9 36.1 41.3 46.4 21.7 1.69 5.63 "3 16.9 22.5 28.1 33.8 39.4 4S.O 50.6 19,9 »3/i6 1.83 6.09 12.2 18.3 24.4 30.5 36.6 42.7 48.8 54.9 18.4 ^. 1.97 6.56 13. 1 19.7 26.3 32.8 39.4 45.9 52.5 59.1 17.I »5/i6 2.II 7.03 14.1 21. 1 28.1 35.2 42.2 49.2 56.3 63.3 15.9 I 2.25 7.50 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 14.9 inches. 5/16 H 7/16 9/x6 "As H '3/16 H «s/x6 sq. in. .594 .742 .891 1.04 1. 19 1-34 1.48 1.67 1.78 .08 1-9 2.2 2.3 lbs. 1.98 2.47 2.97 3.46 396 4.45 4.95 5-44 5-94 6.43 6.93 7.42 7.92 lbs. 3.96 4.95 5.94 6.93 7.92 8.91 9.90 10.9 II. 9 12.9 13.9 14.8 15.8 lbs. 5-94 7.42 8.91 10.4 II. 9 134 14.8 16.3 17.8 19.5 20.8 22.3 23.8 lbs. 7.92 9.90 II.9 139 15.8 17.8 19.8 21.8 23.8 25-7 27.7 29.7 31.7 lbs. 9.90 12.4 14.8 '73 19 i 22.3 24.7 27.2 29.7 32.2 34-6 37.1 39.6 lbs. II. 9 14.8 17.8 20.8 23.8 26.7 29.7 32.7 35.6 38.6 41.6 44.5 47.5 lbs. 13-9 17.3 20.8 24.2 27-7 31.2 34-6 38.1 41.6 45 -o 48.5 5".9 55-4 lbs. 15.8 19.8 23.8 27.7 3J.7 35-6 39.6 43-5 47.5 51.5 55-4 59.4 63.3 lbs. 17.8 22.3 26.7 312 35-6 40.1 43.5 49.0 53-4 57.9 62.3 66.8 71.3 feet. 56.6 45.3 37.7 32.3 28.3 25.2 22.6 20.6 18.9 17.4 16.2 15.1 14.2 FLAT BAR IRON. 231 Weight of Flat Bar Iron. 2% INCHES Wide. ' Length in ] Fekt. T *t Thick- Sbct. Area. Len^h to weigh KESS. X a 3 4 5 6 7 8 9 I cwt. iaches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. % .625 2,08 4.17 6.25 8.33 10.4 12.5 14.6 16.7 18.8 53.8 5/16 .781 2.60 5.21 7.81 10.4 13.0 15.6 18.2 20.8 23.4 28.1 43.0 H .938 3.>3 6.25 9.38 '2-5 15.6 18.8 21.9 25.0 35.8 7/16 1.09 3.65 7.29 10.9 14.6 18.2 21.9 25.5 29.2 32.8 30.7 'A 1.25 4.17 8.33 12.5 16.7 20.8 25.0 29.2 33.3 37.5 26.9 9/t6 I.4I 4.69 9.38 14. 1 18.8 23.4 28.1 32.8 37-5 42.2 23.9 H 1.56 5.21 10.4 15.6 20.8 26.0 3».3 36.5 41-7 46.9 21.5 "A6 1.72 S-73 II.5 17.2 22.9 28.6 34.4 40.1 45.8 51.6 19.6 H I.S8 6.25 12.5 18.6 25.0 31.3 37.5 43.8 50.0 56.3 18.0 n/i6 2.03 6.77 13.5 20.3 27.1 33.8 40.6 47.4 54.2 60.9 16.5 '^, 2.19 7.29 14.6 21.9 29.2 36.5 43.7 51.0 58.3 65.7 15.4 iS/i6 2-34 7.81 15.6 23.4, 31.3 39.0 46.9 54.7 62.5 70.3 14-3 I 1 2.50 8.33 16.7 25.0 33-3 41.7 50.0 58.3 66.7 75.0 13.4 2^ INCHES Wide. tnrbrf. sq. in. .656 lbs. 2.19 lbs. 4.38 lbs. 6.56 lbs. 8.75 lbs. 10.9 lbs. 13.1 lbs. 15.3 lbs. 17.5 lbs. 19.7 feet. 5>.2 J4, 5/.6 .820 2.73 5-47 8.20 10.9 13.7 16.4 19. 1 21.9 24.6 41.0 H .984 3.28 6.56 9.84 13.1 16.4 19.7 23.0 26.2 29.5 34.2 7/16 I.I5 3. 8 J 7.66 II.5 15.3 19. 1 23.0 26.8 30.6 34.4 29.3 }i I.3I 4.38 8.75 13.1 17.5 21.9 26.3 30.6 3S-0 39.4 25.6 ,/;6 1.48 4.92 9.84 14.8 19.7 24.6 29.5 34-5 39-4 44.3 22.8 ^, 1.64 5-47 10.9 16.4 21.9 27.3 32.8 38.3 43.8 49.2 20.5 "A« I.8I 6.02 12.0 18. 1 24.1 30.2 1 36.1 42.1 48.1 54.1 18.6 H 1.97 6.56 13. 1 19.7 26.3 32.8 1 39.4 45.9 52.5 59.1 17. 1 •3/16 2.13 7.11 14.2 21.3 28.4 35-5 1 42.7 49.8 56.9 64.0 15.8 ^. 2.30 7.66 15-3 23.0 30.6 38.3 ! 45.9 53-6 61.3 68.9. 14.7 »s/lfi 2.46 8.20 16.4 24.6 32.8 41.0 I 49.2 57.4 65.6 73.8 13.7 I 2.63 8.75 17.5 26.3 35.0 43.8 52.5 61,3 70.0 78.8 12.8 2^ INCHES Wide. 1 indies. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X .688 2.29 4.58 6.87 9.17 II.5 13.8 15.0 18.3 20.6 48.9 5/x6 .859 2.86 5-73 8.59 II. 5 14.3 17.2 20.1 22.9 25.8 39.1 H 1.03 3-44 6.88 10.3 13.8 17.2 20.6 24.1 27.5 309 32.8 7/16 1.20 4.01 8.02 12.0 16.0 20.1 24.1 28.1 32.1 36.1 27.9 }^ 1-38 4.58 9.17 13.8 18.3 22.9 27.5 32.1 36.7 41.3 24.4 9/16 1.55 5.16 IO-3 15.5 20.6 25.8 30.9 36.1 41.3 46.4 21.7 H 1.72 5.73 11.5 17.2 22.9 28.6 34.4 40.1 45.8 51.6 *9-5 "/t6 1.89 6.30 12.6 18.9 25.2 31.5 37.8 44.1 50.4 56.7 17.8 M 2.06 6.88 138 20.6 27.5 34.4 41.3 48.1 55.0 61.9 16.3 n/16 2.23 7.45 14.9 22.3 29.8 37.2 44.7 52.1 59.6 67.0 15.0 Ji 2.41 8.02 16.0 24.1 32.1 40.1 48.1 56.1 64.2 72.2 14.0 *S/x6 2.58 8.59 17.2 25.8 34-4 430 51.6 60.1 68.8 77.3 13.0 1 2.75 9.17 18.3 27.5 36.7 45.8 55.0 64.2 73.3 82.5 12.2 232 WEIGHT OF METALS. Weight of Flat Bar Iron. 2% INCHES Wide. Length in ] Feet. Thick- ness. Sect. Area. Length to weigh z 2 3 4 5 6 7 8 9 X cwt. lAches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. %, .719 2.40 4.79 7.19 9.58 12.0 14.4 16.8 19.2 21.6 46.7 5/16 .898 3.00 6.cx> 9.00 12.0 15.0 18.0 21.0 24.0 27.0 37.4 H 1.08 3.59 7.19 10.8 14.4 iS.o 21.6 25.2 28.8 32.3 312 7/16 1.26 4.19 8.39 12.6 16.8 21.0 25.2 29.4 33-5 37.7 26.7 'A 1.44 4-79 9.58 14.4 19.2 24.0 28.8 33-5 38.3 43.1 23.4 9/16 1.62 5-39 10.8 16.2 21.6 27.0 32.3 37.7 ^3-' 48.5 20.8 ^, 1.80 5-99 12.0 18.0 24.0 30-0 36.0 42.0 48.0 54,0 18.7 "A6 1.98 6.59 13.2 19.8 26.4 33.0 40.5 46.1 52.7 59.3 17.0 H 2.16 7.19 14.4 21.6 28.8 36.0 43.1 50.3 57.5 64.7 1 15.6 t' 2.34 7.79 ^H 23.4 311 39.0 46.7 54.5 62.3 70.1 14.4 2.52 8.39 16.8 25.2 33.5 42.0 503 58.7 67.1 75-5 13.4 »5/i6 2.70 8.98 18.0 27.0 35.9 45.0 53.9 62.9 71.9 80.9 12.4 I 2.88 9.58 19.2 28.8 38.3 48.0 57-5 67.1 76.7 86.3 II.7 3 INCHES Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. 1 lbs. lbs. lbs. lbs. feet. % .750 2.50 5.00 7.50 lO.O 12.5 15.0 17.5 20.0 22.5 44.8 s/i6 .938 3-13 6.25 9.38 I2.S '^Z 18.8 21.9 25.0 28.1 35.8 H 1. 13 3.75 7.50 II-3 15.0 18.8 22.5 26.3 30.0 33.8 29.9 7/i6 I-3I 438 8.75 131 17.5 21.9 26.3 30.6 350 39.4 25.6 'A 1.50 5.00 lO.O 15.0 20.0 25.0 30.0 35.0 40.0 45.0 22.4 9/16 1.69 5-63 "3 16.9 22.5 28.2 33.8 39.4 450 50.6 .19.9 H 1.88 6.25 12.5 18.8 25.0 31-3 37.5 43.8 50,0 56.3 17.9 r 2.06 6.88 13.8 20.6 27.5 34.4 41.3 48.1 55.0 61.9 16.3 2.25 7.50 15.0 22.5 30.0 37-5 45.0 52.5 60.0 67.5 14.9 T 2.44 8.13 16.3 24.4 32.5 40.7 48.8 56.9 65.0 73-' 13.8 2.63 8.75 '7-5 26.3 35.0 43.8 52.5 61.3 70.0 78.8 12.8 ^5/16 1 2.81 9.38 18.8 28.1 37-5 46.9 56.3 65.6 75.0 84.4 12.0 I 1 1 3-0O 10.0 20.0 300 40.0 50.0 60.0 70.0 80.0 90.0 II. 2 3X inches Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. fecL X .813 2.71 5.42 8.13 10.8 13.6 16.3 19.0 21.7 24.4 41.3 5/16 1.02 3.39 6.77 10.2 13.5 16.9 20.3 23.7 27.1 30.5 33.1 H 1.22 4.06 8.13 12.2 16.3 20.3 24.4 28.4 32.5 36.6 27.5 7/16 1.42 4.74 9.48 14.2 19.0 237 28.4 33.2 37.9 4?/Z 23.6 yi 1.63 5.42 10.8 16.3 21.7 27.1 32.5 37.9 43-3 48.8 20.7 9/.« 1.83 6.09 12.2 18.3 24.4 305 36.6 42.7 48.7 54.8 18.4 H 2.03 6.77 13.5 20.3 27.1 33.9 40.6 47.4 54.2 60.9 16.5 r 2.23 7-45 14.9 22.3 29.8 37.2 44.7 52.1 59.6 67.0 15.0 2.44 8.13 16.3 24.4 32.5 40.6 48.8 56.9 65.0 731 13.7 r 2.64 8.80 17.6 26.4 35.2 44.0 52.8 61.6 70.4 79.2 12.7 2.84 9.48 19.0 28.4 37.9 47.4 56.9 66.4 75.8 85.3 II.8 »5/i6 3.05 10.2 20.3 30. 5 40.6 50.8 60.9 71. 1 81.2 91.4 II.O I 325 10.8 21.7 32. 5 43.3 54.2 65.0 75.8 86.7 97.5 10.3 FLAT BAR IRON. 233 Weight of Flat Bar Iron. S}4 INCHES Wide. 1 Lbngth in Fbbt. Thick- 1 Sect. Area. Lens|th to weigh I a 3 4 5 6 7 8 9 I CWL 1 inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. ><■ .875 2.92 5.83 8.75 II.7 14.6 17.5 2G.4 233 26. J 38.4 s/16 1.09 3-^5 7.29 10.9 14.6 18.2 21.9 25.5 29.2 32.8 30.7 H I-3I 4.38 8.75 13. 1 17.5 21.9 26.3 30.6 35.0 39.4 25.6 7/16 1.53 5.10 10.2 15.3 20.4 25.5 30.6 35-Z 40.8 45-9 21.9 }i 1.75 5.83 II. 7 17.5 22.3 29.2 35.0 40.8 46.7 52.5 19.2 S/.6 ' 1.97 6.56 13. 1 19.7 26.3 32.8 39.4 45-9 55-5 5^1 17. 1 H 2.19 7.29 14.6 21.9 29.2 36.5 43.7 51.0 58.3 65.6 15-4 T 2.41 8.02 16.0 24.1 32.1 40.1 48.1 56.1 64.2 72.2 14.0 2.63 8.75 «7.5 26.3 350 43.8 52.5 61.3 70.0 78.8 12.8 '3/16 2.84 9.48 19.0 28.4 37.9 47.4 56.9 66.4 75.8 85.3 11.9 H 3.06 10.2 20.4 306 40.8 51.0 61.2 71.5 81.6 91.9 II. «5/i6 3.28 10.9 21.9 32.8 43.8 54.7 65.6 76.6 87.5 98.4 10.2 I ' 3.50 II.7 23.3 35.0 46.7 58.3 70.0 81.7 93.3 105.0 9.60 3j^ INCHES Wide. t inches. 5/16 H 7/16 9/16 H "1x6 H «j/i6 »5/i6 I sq. m. .938 I.I7 I.4I 1.64 1.88 2. II 2.34 2.58 2.81 3.05 3-28 3.52 3-75 lbs. lbs. 3.13 3.91 4.69 6.25 6.25 7.81 9.38 10.9 12.5 7.03 7.81 8.59 9.38 14. 1 15.6 17.2 18.8 10.2 20.3 10.9 II.7 12.5 21.9 23.4 25.0 lbs. lbs. lbs. lbs. lbs. lbs. feet. 12.5 15.6 18.8 21.9 25.0 28.1 35-^ 15.6 19.5 23.4 27.3 32.8 3^.3 35.2 28.7 18.8 234 28.1 37.5 42.2 23.9 21.9 27.3 32.8 3^-2 43.7 49.2 20.5 25.0 3».3 37.5 43.8 50.0 56.3 17.9 28.1 35.3 42.2 49.2 56.3 63.3 15.9 3>.2 39.1 46.9 54.7 62.5 70.3 '4.3 34.4 43.0 51.6 60.2 68.8 27-3 13.0 37.5 46.9 56.3 65.6 75.0 84.4 12.0 40.6 50.8 60.9 7I.I 81.3 91.4 II.O 43.8 54.7 65.6 76.6 87.5 98.4 10.2 46.9 58.6 70.3 82.0 93-7 105.5 2-5^ 50.0 62.5 75.0 87.5 100. 112.5 8.96 4 INCHES Wide. inchrt. iq. in. lbs. lbs. lbs. lbs. lbs. lbs. lU. lbs. lbs. feet. '^ 1 I.OO 3.33 6.67 10. 13.3 16.7 20.0 23.3 26.7 30.0 33.6 5/16 ! H 1.25 4.17 8.33 12.5 16.7 20.8 25.0 29.2 33.3 37.5 26.9 1.50 5.00 lO.O 15.0 20.0 25.0 30.0 35.0 40.0 45.0 22.4 7/16 1-75 5.83 II.7 17.5 23-3 29.2 35.0 40.8 46.7 52.5 19.2 H 1 2.00 6.67 13.3 20.0 26.7 33.3 40.0 46.7 53-3 60.0 i6.8 9/16 1 2.25 7.50 15.0 22.5 30.0 37.5 45.0 5;- 5 60.0 67.5 14.9 H 2.50 8.33 16.7 25.0 33-3 41.7 50.0 58.3 66.7 75.0 13.4 "/«6 2.75 9.17 18.3 27.5 36.7 45.8 55.0 64.2 73.3 82.5 12.2 1 H ' ' 300 lO.O 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 II. 2 13/.6 1 1 3.25 10.8 21.7 32.5 43-3 54.2 65.0 Z5-» 86.7 97.5 10.3 ^, 3.50 II.7 23.3 35.0 46.7 58.4 70.0 81.7 93.3 105.0 9.60 ' «5/.6 , 3-75 12.5 25.0 37.5 50.0 62.5 75.0 87.5 lOO.O II2.5 8.96 I 4.00 13.3 26.7 40.0 53.3 66.7 80.0 93.3 106.7 120.0 8.40 234 WEIGHT OF METALS. Weight of Flat Bar Iron. 4X INCHES Wide. Length in Feet. Thick- Sect. Area. Len^h to weigh ness. I 2 3 4 5 6 7 8 9 I OWL inches. 1 sq. ixu lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet V 1.06 3^54 7.08 10.6 14.2 17.7 21.3 24.8 28.3 31.9 31.6 5/16 1.33 4.43 8.85 13.3 17.7 22.1 26.6 3I-0 35.4 39.8 25.3 H ^•59 5.31 10.6 »5-9 21.3 26.6 31.9 37.2 42.5 47.8 21. 1 7/16 1.85 6.20 12.4 18.6 24.8 31.0 37.2 43-4 49.6 55.8 18. 1 ^ 2.13 7.08 14.2 21.3 28.3 35.4 42.5 49.6 56.7 63.8 15.8 9/16 2.39 ni 15.9 23.9 31.9 39.8 47.8 55.8 63.7 71.7 14. 1 1 H, 2.66 17.7 26.6 35.4 44.3 53-1 62.0 70.8 79.7 12.7 t' 2.92 9.74 19.5 29.2 39.0 48.7 58.4 68.2 77.9 87.7 II.5 3-19 10.6 21.3 31.9 42.5 53-1 63.8 74-4 85.0 95.6 10.5 «3/i6 3.45 11.5 23.0 34.5 46.0 57.6 69.1 80.6 92.1 103.6 9-9 n 3.72 12.4 24.8 37.2 49.6 62.0 74.4 86.8 99.2 III. 6 9.0 ^5/16 3.98 13.3 26.6 39.« 53.1 66.4 79.7 93-0 106.2 "95 8.4 I 4.25 14.2 28.3 42.5 56.7 70.8 85.0 99.2 "33 127.5 1 7.9 4}4 iN'CHEs Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. i lbs. lbs. lbs. lbs. feet ^ I.I3 3.75 7-5, "•3 15.0 18.8 22.5 26.3 30-0 33.8 29.9 5/16 I.4I 4.69 9.38 14. 1 18.8 \l\ 28.1 32.8 37.5 42.2 23.9 ^ 1.69 5.63 "3 16.9 22.5 33.8 39-4 45.0 50.6 19.9 7/16 1.97 6.56 131 19.7 26.3 32.8 , 39-4 45-9 52.5 59.1 17. 1 H 2.25 7.50 15.0 22.5 30.0 37.5 1 45.0 52.5 60.0 67.5 14.9 9/16 2.53 8.44 16.9 25.3 33.8 42.2 50.6 59.1 67.5 75.9 13-3 H 2.81 9.38 18.8 28.1 37.5 46.9 , 56.3 65.6 75.0 84.4 12.0 r 3.09 10.3 20.6 • 30.9 41.3 51.6 61.9 72.2 82.5 92.8 10.9 3-38 "3 22.5 33.8 45.0 56.3 67.5 78.8 90.0 IOI.3 9-95 t' 3.66 12.2 24.4 36.6 48.8 60.9 ' 721 85.3 97.5 109.7 9.19 3-94 13.1 •26.3 39-4 52.5 65.6 78.8 91.9 105.0 118. 1 ».53 »s/i6 4.22 14. 1 28.1 42.2 56.3 70.3 84.4 98.4 II2.5 126.6 7.96 I 4.50 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0 135.0 7.46 ^}( INCHES Wide, inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet % 1. 19 3.96 7.92 II.9 15.8 19.8 23.8 27.7 31.7 35.6 28.3 she 1.48 4.95 9.90 14.8 19.8 24.8 29.7 34.6 39.6 44.4 22.6 H 1.78 5-94 11.9 17.8 23.8 29.7 35-6 41.6 47.5 53.4 18.9 7/16 2.08 6.93 '3-2 20.8 27.7 34-7 41.6 48.5 55-4 62.3 16.2 ^ 2.38 7.92 15.8 23.8 31.7 39.6 47.5 55.4 63.3 71.3 14.2 9/.6 2.67 8.91 17.8 26.7 35.6 44.6 53-4 62.3 71.3 80.2 12.6 ^, 2.97 9.90 19.8 29.7 39.6 49-5 1 59-4 69-3 79.2 89.1 "3 r 3.27 10.9 "S 32.7 43.5 54.5 ' 65.3 76.2 87.1 98.0 10.3 3.56 1 1.9 23.8 35.6 47. 5 59.4 71.3 83.1 95.0 106.9 9-4 «3/i6 3.86 12.9 25.7 38.6 51.5 64.3 77.2 90.1 102.9 II5.8 8.7 ^> 4.16 13-9 27.7 41.6 55.4 69.3 83.1 97.0 no. 8 124.7 8.1 '5/16 4.45 14.8 29.7 44.5 59.4 74.2 89.1 103.Q 118.8 133-6 7.5 I 4.75 15.8 31-7 47.5 633 79.2 95.0 1 10.8 126.7 142.5 7.1 FLAT BAR IRON. 235 Weight of Flat Bar Iron. 5 INCHES Wide. Length in ] Feet. Thick- ness. Sect. ASEA. Len^h to weigh z 2 3 4 5 6 7 8 9 I cwt. ipches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. u 1.25 4.17 8.33 12.5 16.7 20.8 20.9 25.0 29.2 33.3 37.5 26.9 S/>6 l!88 5.21 10.4 15.6 26.1 3>.3 36. J 43.8 41.7 46.9 21.5 H 6.25 12.5 18.8 25.0 3>.3 37-5 50.0 56.3 17.9 7/16 2.19 7.29 14.6 21.9 29.2 36.5 43.8 5i-° 58.3 65.6 15.4 a 2.50 8-33 16.7 25.0 33.3 41.7 50.0 58.3 66.7 75-0 13.4 9/16 2.81 9.38 18.8 28.1 37.5 46.9 56.3 65.6 75.0 84.4 12.0 ><, 3.13 10.4 20.8 313 41.7 52.1 62.5 72.9 83.3 93-8 10.8 r 3-44 11.5 22.9 34.4 45.8 57.3 68.8 80.2 91.7 103. 1 9-77 3.75 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100. 112.5 8.96 't 4.06 13.5 27.1 40.6 54.2 67.7 81.3 94.8 108.3 121. 9 8.27 4.38 14.6 29.2 43.8 58.3 72.9 87.5 102. 1 II6.7 '31.3 7.68 «5/i6 4.69 15.6 31.3 46.9 62.5 78.1 93.8 109.4 125.0 140.6 7.17 I 5.00 16.7 33.3 50.0 66.7 83.3 lOO.O 116. 7 133.3 150.0 6.72 5^ INCHES Wide. inchfs sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet V I-3I 4.38 8.75 13-1 17.5 21.9 26.3 32.8 30.6 35.0 39-4 25.6 s/16 1.64 §•47 10.9 16.4 21.9 27.3 38.3 43.8 49.2 20.5 H 1.97 6.56 "3-1 19.7 26.3 32.8 39-4 1 45-9 52.5 59.1 I7.I 7/16 2.30 7.66 15.3 23.0 30.6 38.3 45-9 53.6 61.3 68.9 14.6 >i 2.63 8.75 17.5 26.3 35-0 43-8 52.5 61.3 70.0 78.8 12.8 9/16 2.95 9.84 19.7 29.5 39.4 43.8 49.2 59.1 68.9 78.8 88.6 II.4 H 3.28 10.9 21.9 32.8 54-7 65.6 76.6 87.5 98.4 I0.3 r 3.61 12.0 24.1 36.1 48.1 60.2 72.2 84.2 96.3 108.3 9.31 3.94 13. 1 26.3 39.4 52.5 65.6 78.8 91.9 105.0 118.1 8.55 »3/i6 4.27 14.2 28.4 42.7 56.9 71. 1 85.3 99.5 113.7 128.0 7.88 H 4.59 »5.3 30.6 45.9 61.3 76.6 91.9 107.2 122.5 137.8 7-3* ^s/i6 4.92 16.4 32.8 49.2 65.6 82.0 98.4 114.8 131. 3 147.7 6.S3 I 5.25 17.5 35.0 52.5 70.0 87.5 105.0 122.5 140.0 157.S 6.40 $}4 INCHES Wide. indies. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. 'X 1.38 4.58 9.17 13.8 18.3 22.9 27.5 32.1 36.7 41.3 24.5 S/t6 1.72 m 11. g 17.2 22.9 28.6 34.4 40.1 45.8 51.6 '95 H 2.06 13.8 20.6 27.5 34.4 4i-3 48.1 55.0 61.9 16.4 7/16 2.41 8.02 16.0 24.1 32.1 40.1 48.1 56.1 64.2 72.2 14.0 >i 2.75 9-17 18.3 27.5 36.7 45.8 55-0 64.2 73.3 82.5 12.2 9/16 3.09 W.3 20.6 30.9 41.3 51.6 61.9 72.2 82.5 92.8 10.9 H 3.44 11.5 22.9 34.4 45.8 57.3 68.8 80.2 91.7 103. 1 9.77 "ft6 3.78 12.6 25.2 37.8 50.4 63.0 68.8 75.6 88.2 100.8 113.4 8.88 H 4.13 13.8 27.5 41.3 55.0 82.5 96.3 IIO.O 123.8 8.14 T 1 4-47 14.9 29.8 44.7 59.6 74.5 89.4 104.3 119. 2 134. 1 7.52 4.81 16.0 3^.1 48.1 ^l 80.2 96.3 H2.3 128.3 144.4 6.98 »5/i6 5.16 17.2 34.4 51.6 68.8 85.9 103. 1 120.3 137.5 154.7 6.52 ' 5-50 "8.3 36.7 55.0 73-3 91.6 IIO.O 1 128.4 146.7 165.0 6.11 236 WEIGHT OF METALS. Weight of Flat Bar Iron. $}( INCHES Wide. Lbngth in Fbbt. T *.!_ Thick- ness. Sbct, Arba. Length to weigh z 2 3 4 5 6 7 . 8 9 X cwt. inches. 1 sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. m: 1.44 4.79 9.58 14.4 19.2 24.0 28.8 33.5 38.3 43-1 23.4 s/16 1.80 5-99 12.0 i8.o 24.0 30.0 35.9 41.9 47.9 53.9 18.7 H 2.16 7.19 14.4 21.6 28.8 35.9 43.1 50.3 57.5 64.7 15.6 7/j« 2.52 8.39 16.8 25.2 33.5 38.3 41.9 50.3 58.7 67.1 75-5 134 H 2.88 9.58 19.2 28.8 47.9 57.5 67.1 76.7 86.3 II.7 9/«« 3.23 10.8 21.6 32.3 43.' 48.0 53.9 64.7 75.5 86.2 97.0 10.4 H 3.59 12.0 24.0 36.0 60.0 71.9 83.9 95.8 107.8 9-35 r 3.95 13.2 ^t-i 39.5 52.7 65.9 79.1 92.2 105.4 1 18.6 8.50 4.31 14.4 28.8 43.1 57.5 71.9 86.3 100.6 115.0 129.4 7.79 r 4.67 15.6 31.2 46.7 62.3 77.9 93-4 109.0 124.6 140.2 7.19 5.03 16.8 33.5 50.3 67.0 83.9 ic».7 1 17.4 134.2 150.9 6.68 '5/x6 5.39 18.0 35.9 53-9 71.9 89.8 107.8 125.8 H3-7 161. 7 6.22 I 5.75 19.2 38.3 57.5 76.7 95.8 115.0 "34.2 153.3 172.5 5.83 6 INCHES Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. H 1.50 5.00 lO.O 15.0 20.0 25.0 30.0 35.0 40.0 45.0 22.4 s/x6 1.88 6.25 12.5 18.8 25.0 31.8 37.5 43.8 50.0 56.3 17.9 H 2.25 7.50 15.0 22.5 30.0 37-5 45.0 52.5 60.0 67.5 14.9 7/16 2.63 8.75 17.5 26.3 350 43.8 52.5 61.3 70.0 80.0 78.8 12.8 H 3.00 10.0 20.0 30.0 40.0 50.0 60.0 70.0 90.0 II. 2 9/x6 3.38 "3 22.5 33.8 45.0 56.3 67.5 78.8 90.0 101.3 lO.O H 3-75 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0 112.5 8.96 r 4.>3 4.50 13.8 15.0 27.5 30.0 41-3 450 60.0 68.8 75.0 82.5 90.0 96.3 105.0 HO.O 120.0 123.7 135.0 8.15 7.47 t 4.88 16.3 32.5 48.8 65.0 81.3 97.5 "37 1300 146.3 6.90 5.25 '7-5 18.8 35.0 52.5 70.0 87.$ 105.0 122.5 140.0 157.5 6.40 >5/i6 5.63 37.5 56.3 75.0 93.8 II2.5 131.3 150.0 168.7 5-97 I 6.00 20.0 40.0 60.0 80.0 lOO.O 120.0 140.0 160.0 180.0 S.60 6% INCHES Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. % 1.63 5-42 10.8 16.3 21.7 27.2 32.5 37.9 43.3 49.0 20.7 5/x6 2.03 6.77 13.5 20.3 27.1 33-9 40.6 47.4 54.2 60.9 16.5 H 2.44 8.13 16.3 24.4 32.5 40.6 48.8 56.9 65.0 73.1 7/x6 2. 84 9.47 18.9 2S.4 37.9 47.4 56.8 66.3 75.8 85.2 14.8 }i 3.25 10.8 21.7 32.5 43.3 54.2 65.0 75.8 86.7 97.5 10.3 9/x6 3.66 12.2 24.4 36.6 48.8 60.9 1 73- ' 85.3 97-5 109.7 9.20 >^. 4.06 13.5 27.1 40.6 54.2 67.7 81.3 94.8 108.3 121.9 8.27 "/i6 4-47 14.9 29.8 44.7 59.6 74.5 89.4 104.3 1 13.8 119.2 134.1 6.89 H 4.98 16.3 32.5 48.8 65.0 81.3 97.5 130.0 146.3 '3/x6 5.28 17.6 35-2 52.8 70.4 88.0 105.6 123.2 140.8 158.4 6.36 H 5.68 19.0 37-9 56.9 75.8 81.3 94.8 1 13.8 132.7 151.7 170.6 5-91 '5/x6 6.09 20.3 40.6 60.9 101.6 121.9 142.8 162.5 182.8 5.51 I 6.50 21.7 43.3 65.0 86.7 108.3 130.0 1 I5I.7 173-3 195.0 5.29 FLAT BAR IRON. 237 Weight of Flat Bar Iron. 7 INXHES Wide. Length in Feet. Thick- ness. Sect. Area. Length to weigh X 2 3 4 5 6 7 8 9 I CWL * inches. sq. m. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X 1.75 583 II.7 17.S 23.3 29.2 35.0 40.8 46.7 52.5 19.2 s/i6 2.19 7.29 14.6 21.9 29.2 3^-5 43.8 51.0 58.3 65.6 15.4 H 2.63 8.75 17.5 26.3 35-2 43-8 52.5 61.3 70.0 78.8 12.8 I ?/«« 3-o6 10.2 20.4 30.6 40.8 5?/° 61.3 7^-5 81.7 91.9 II.O 1 H 3.50 II.7 23.3 350 '46.7 58.3 70.0 81.7 93-3 105.0 9.60 9/16 3-94 13. 1 26.3 39.4 5;^- 5 65.6 78.8 91.9 105.0 I18.I 8.53 H 4.38 14.6 29.2 43.8 58.3 72.9 87.5 102. 1 1 16. 7 131.3 7.68 r 4-81 5.25 16.0 17.5 32.1 35.0 48. 1 52.5 64.2 70.0 80.2 87.5 96.3 105.0 112.3 122.5 128.3 140.0 144-4 157.5 6.98 6.40 «3/i6 5.69 19.0 37-9 56.9 75.8 95.0 II3.8 132.7 151.7 170.6 591 H 6.13 20.4 40.8 61.3 81.7 102. 1 122.5 142.9 163.3 183.8 5-49 «5/i6 6.56 21.9 43.8 65.6 87.5 109.4 131-3 '53- » 175.0 196.9 5.12 I 7.00 23.3 46.7 70.0 93.3 1 16. 7 140.0 163.3 186.7 210.0 4.80 y}^ INCHES Wide. baches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X 1.88 6.25 12.5 18.8 25.0 31.3 37-5 43.8 50.0 56.3 17.9 s/16 2.34 7.81 15.6 23.4 31.3 39. > 46.9 54.7 62.5 70.3 14-3 H i 2.81 9.38 18.8 28.1 37-5 46.9 56.3 65.6 75.0 84.4 II.9 7/16 3-28 10.9 21.9 32.8 43.8 54.7 65.6 76.6 87.5 98.4 10.2 >i 3.75 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100. H2.5 8.96 9/16 4.22 14. 1 28.1 42.2 56.3 70.3 84.4 98.4 II2.5 126.6 7.96 H 4.69 15.6 3Jf.3 46.9 62.5 78.1 93.8 109.4 125.0 140.6 7.17 "/16 5.16 "^^'l 34-4 51.6 68.8 85.9 103. 1 120.3 137.5 154.7 6.52 H 5.63 18.8 37.5 56.3 75.0 93.8 112.5 131-3 150.0 168.8 5.97 T 6.09 20.3 40.6 60.9 81.3 I0I.6 121. 9 142.2 162.5 182.8 5-51 6.56 21.9 43.8 65.6 87.5 109.4 131.3 153.1 175.0 196.9 5.12 »5/i6 7-03 23.4 46.9 70.3 93.8 117.2 140.6 164. 1 187.5 210.9 4.78 I 7.50 25.0 50.0 75.0 100. 125.0 150.0 175.0 200.0 225.0 4.48 8 INCHES Wide. inches. I sq. in. lbs. lbs. lbs. lbs lbs. lbs. lbs. lbs. lbs. feet. X 1 2.00 6.67 13.3 20.0 26.7 33-3 40.0 46.7 53-3 60.0 16.8 s/16 2.50 8.33 16.7 25.0 33.3 41.7 50.0 58.3 66.7 75.0 13.4 H ' 300 10. 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 II. 2 7/16 3-50 II.7 233 35.0 46.7 58.3 70.0 81.7 93.3 105.0 0.60 1 ^ 4.00 13.3 26.7 40.0 53-3 66.7 80.0 93-3 106.7 120.0 8.40 9/16 4.50 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0 135.0 7.47 ^, 5.00 16.7 33.3 co.o 66.7 83.3 lOO.O 116.7 133.3 150.0 6.72 t' 5.50 18.3 36.7 55.0 73.3 91.7 IIO.O 128.3 146.7 165.0 6.11 6.00 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0 5.60 »3/i6 6.50 21.7 43.3 65.0 86.7 108.3 130.0 151. 7 173.3 195.0 5-i7 », 7.00 23-3 46.7 70.0 93-3 116.7 140.0 163.3 186.7 2IOkO 4.80 ^5fi6 7.50 25.0 50.0 75.0 100.0 125.0 150.0 175.0 200.0 225.0 4.48 1 8.00 26.7 53.3 80.0 106.7 133.3 160.0 186.7 213.3 240.0 4.20 238 WEIGHT OF METALS. Weight of Flat Bar Iron. 9 INCHES Wide. Thicic- NESS. Sect. Area. Length in Fbbt. Length to weigh I cwt. z 1 3 4 5 6 7 8 9 inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. 5/16 H 7/16 2.25 2.81 3.38 3-94 4,50 7-50 9.38 "3 13. 1 15.0 15.0 18.8 22.5 26.3 30.0 22.5 28.1 33-8 39-4 45-0 30.0 37-5 45.0 60.0 37.5 46.9 tl 65.6 75-0 45.0 56.3 Hi 78.8 • 90.0 52.5 65.6 78.8 91.9 105.0 60.0 75.0 90.0 105.0 120.0 67.5 84.4 IOI.3 118. 1 135.0 14.9 II.9 10.0 8.53 7.47 9/16 5.06 5.63 6.19 6.75 16.9 18.8 20.6 22.5 33.8 37-5 41-3 45.0 50.6 56.3 61.9 67.5 67.5 75.0 82.5 90.0 84.4 93-8 103. 1 112.5 101.3 112.5 123.8 135-0 118. 1 I3I.3 144.4 157.5 1350 150.0 165.0 180.0 151.9 168.8 185.6 202.5 6.64 5-97 5-43 4.98 »5/i6 I 8.44 9.00 24.4 26.3 28.1 30.0 48.8 56.3 60.0 73.1 78.8 84.4 90.0 97-5 105.0 112.5 120.0 121. 9 i3>-3 140.6 150.0 146.3 104.5 168.8 180.0 170.6 183.8 196.9 210.0 195.0 210.0 225.0 240.0 219,4 236.3 253-1 270.0 4.59 4.26 3.98 3.73 10 INCHES Wide. inches. ' 1 sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. ^ 2.50 8.33 16.7 25.0 33-3 41.7 50.0 58.3 66.7 75.0 13-4 S/.6 3.13 10.4 20.8 31.3 41.7 52.1 62.5 72.9 83.3 93.8 10.7 ^ 3.75 12.5 25.0 37-5 50.0 62.5 75.0 87-5 100.0 II2.5 8.96 7/16 4-38 14.6 29.2 43-8 58.3 72.9 87-5 102. 1 116.7 I3I-3 7.68 }i 5.00 16.7 33-3 50.0 66.7 83.3 lOO.O 116.7 133.3 150.0 6.72 9/16 5.63 18.8 37.5 56.3 75.0 93.8 II2.5 13I-3 150.0 168.8 5-97 H 6.25 20.8 ^^■7 62.5 83-3 104.2 125.0 145.8 166.7 187.5 4^89 r 6.88 22.9 45.8 68.8 91-7 1 14.6 137-5 160.4 183-3 206.3 7.50 25.0 50.0 75.0 100. 125.0 150.0 175.0 200.0 225.0 4.48 »3/i6 8.13 27.1 54.2 81.3 108.3 135-4 162.5 189.6 216.7 243.8 4.14 '^. 8.75 29.2 58.3 87.5 93-8 1 16. 7 145.8 175.0 ^^'i 233-3 262.5 3.84 ^s/i6 9.40 31-3 62.5 125.0 156.3 187.5 218.8 250.0 281.3 3.58 I 10.0 33-3 66.7 100. 133.3 166.7 200.0 233-3 266.7 300.0 3.36 II INCHES Wide. inches. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. ^ 2-75 9.17 18.3 27.5 36.7 45.8 §5-2 s/16 3-44 11. J 13-8 22.9 34.4 45.8 68.' i 68.8 H 4. "3 27-5 41-3 55.0 82.5 7/16 4.81 16.0 32.1 48.1 64.2 i6.2 96.3 }4 5.50 18.3 36.7 55.0 73.3 91.7 1 10.0 9/16 6.19 20.6 ^^•i 61.9 82.5 103,1 123.8 ^, 6.88 22.9 45-8 68.8 91.8 114.6 "37-5 r 7.56 25.2 50.4 75.6 100.8 126.0 i5>.3 8.25 27-5 55.0 82.5 IIO.O 137.5 165.0 r 8.94 29.8 59-6 89.4 II9.2 149.0 178.8 9.63 32.1 64.2 96.3 128.3 160.4 192.5 *5/i6 10.4 34-4 68.8 103. 1 137.5 171.9 206.3 I 11. 36.7 73-3 1 10.0 146.7 183-3 220.0 lbs. lbs. 64-2 73.3 80.2 91.7 96.3 I IIO.O 1 12. 3 128.3 128.3 146-7 144.4 165.0 160.4 1 183.3 176.5 201.7 192.5 220.0 208.5 238.3 224-6 256.7 240.6 I 275.0 256.7 293.3 lbs. feet. 82.5 12.2 103. 1 123.8 9.77 8.15 144.4 165.0 6.98 6.II 185.6 206.3 226.9 5.43 4-89 4-44 247-5 4.07 268.1 288.8 3.76 3.49 3094 330.0 3.26 3.06 1 SQUARE IRON. 239 Table No. 75.— WEIGHT OP SQUARE IRON. Lkngth in Fbbt. w « SiDB. Sbct. Akba. Lens;th to weigh I a 3 4 5 6 7 8 9 X cwt. laches. aq. b. lbs. lbs. lbs. lU. lbs. lbs. lbs. lU. lbs. feet. X .0156 .052 .104 .156 .208 .260 .313 'l^l .417 .469 2154 3/16 •035' .117 .234 •351 .468 .584 .701 .818 .935 1.05 960.0 }( .0625 .208 .4>7 .625 .833 1.04 1.25 1.46 1.67 1.80 537.6 i/l6 .0977 .326 •^^l .977 1.30 1.68 '•95 2.28 2.60 2.93 343.8 H .141 .469 .938 I.4I 1.58 2.34 2.81 3.28 3.75 4.22 238.3 7/,6 ! .191 .638 1.28 I.9I 2.55 3-19 3.83 4.46 5.10 5.74 176.0 H •25 ■833 1.67 2.50 3.33 4.17 5.00 5.83 6.67 7.50 1344 9/l« .316 1.06 2. II 3- 16 4.22 5.27 6.33 7.38 8.44 9.49 106.3 H •391 1.30 2.60 3.91 5.21 6.51 7.81 9.II 10.4 11.7 85.9 r , -473 1.58 3.J5 473 6.30 7.88 9-45 II. 12.6 14.2 71.0 1.563 1.88 3-75 5.63 7.50 8.80 9.38 "•3 13." 15.0 16.9 59.7 T .661 2.20 4,40 6.61 II. 13-2 '54 16.6 19.8 50.8 .766 2.55 5.10 7.66 10.2 12.8 15.3 17.9 20.4 23.0 43.9 *5/i6 ,879 2.93 5.86 8.79 II.7 14.7 17.6 20.5 234 26.4 38.2 1 I.CX> 3.33 6.67 10.0 13.3 16.7 20.0 233 26.7 30.0 33.6 I »/i6 . 1.13 3.76 7.53 "•3 15.I 18.8 22.6 26.3 30.1 33.9 29.7 IH 1.27 4.22 8.44 12.7 16.9 21. 1 ^5-3 29.5 33.8 38.0 26.5 I 3/16 1 I.4I 4.70 9.40 14. 1 18.8 23.5 28.2 32.9 37.6 42.3 23.8 iH : 1-56 5.21 10.4 15.6 20.8 26.0 31.3 36.5 41.7 46.9 21.5 I 5/16 ' 1.72 5.74 1 1.5 17.2 23.0 28.7 344 40.2 45-9 51.7 19.5 iH 1 1'^ 6.30 12.6 1S.9 25.2 31.5 37.8 44.1 50.4 56.7 17.8 I 7/x6 ' 2.07 6.89 13.8 20.7 27.6 345 4>-3 48.2 55.1 62.0 16.2 1% 1 2.25 7.50 15.0 22.5 30.0 37.5 45-0 52.5 60.0 67.5 14.9 I 9/x6 , 2.44 8.14 16.3 24.4 32.6 40.7 48.8 57.0 65.1 73-2 13-8 IK < 2.64 8.80 17.6 26.4 35-^ 44.0 52.8 61.6 70.4 79.2 12.7 I"/i6 2.88 9.60 19.2 28.8 38.4 48.0 57.6 67.2 76.8 86.4 11.7 I^ 3.06 10.2 20.4 30.6 40.8 51.0 61.3 71.4 81.6 91.9 1 II.O I '3/16 ' 3-29 II. 21.9 32.9 43.8 54.8 65.7 76.7 87.6 98.6 10.2 I^ 3.52 11.7 234 35.2 46.9 58.6 70.3 82.0 93.8 105.5 §■56 H5/16 : 3.75 12.5 25.0 37.5 50.1 62.6 25- » 87.6 100. 1 112.6 8.95 2 . 4.00 '3-3 26.7 40.0 53.3 66.7 80.0 93.3 106.7 120.0 8.40 ^>^ 4.52 15.1 30.1 45.2 60.2 75-3 90.3 105.4 120.0 135.5 7.43 25<^ 5.06 16.9 33.8 50.6 67.1 844 101.3 1 18. 1 135.0 151.9 6.64 2>i 5.64 18.8 37.6 56.4 75.2 94.0 112.8 131-6 150.4 169.2 5.96 2>^ 6.25 20.8 41.7 62.5 833 10.4 125.0 145.8 166.6 187.5 5.38 *>i ,6.89 23.0 45-9 68.9 91.9 114.9 137.8 160.8 183.9 206.7 4-99 2^ • 7.56 25.2 50.4 75.6 100.8 126. 1 151.3 176.5 201.7 226.9 444 2^ S.27 27.6 55.1 82.7 1 10.2 137.8 165.3 192.9 220.4 248.0 4.06 3 9,00 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 3.73 3X } 10.6 35-2 70.4 105.6 140.8 176.0 211. 3 246.5 281.7 316.9 3.17 3>^ 1 12.3 40.8 81.7 122.5 163.3 204.2 245.0 285.8 326.7 367.5 2.73 3^ 14. 1 46.9 93.8 140.6 187.5 234.4 281.3 328.1 375.0 421.9 2.38 4 1 16.0 53.3 106.7 160.0 213-3 266.7 320.0 3730 426.0 1 480.0 2.10 4^ . 18. 1 60.2 120.4 180.6 240.8 301. 1 361.2 421.5 481.7 541.9 1.86 4^ 20.3 67.5 1350 202.5 270.0 337.5 405.0 472.5 540.0 607.5 1.66 4¥ 1 22.6 Z5-2 150.4 225.6 300.8 376.1 451.3 526.5 601.7 676.9 1.49 5 25.0 83.3 166.7 250.0 333-3 416.7 500.0 583.3 666.7 750.0 ».34 ^K 27.6 91.9 183.8 275.6 367.5 459.4 551.3 643.1 735-0 826.9 1. 21 5^ 30.3 100.8 201.7 302.5 403.3 504.2 605.0 705.8 806.7 907.5 I.U 53^ 33.1 1 10.2 220.4 330.6 440.8 551.0 661.3 771.5 881.7 991.8 1.02 6 36.0 120.0 240.0 360.0 480.0 600.0 720.0 840.0 960.0 1080 .933 240 WEIGHT OF METALS. Table No. 76.— WEIGHT OF ROUND IRON. Lbngth in Feet. T -_!_ DiAM. Sect. Area. Length to weig}i z a 3 4 5 6 7 8 9 I cwt. inches. 1 1 sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. feet. X 1 .0123 .041 .082 .123 .164 .368 .205 .245 .286 .327 .368 2738 3/16 .0276 .092 .184 .276 .460 •552 •644 .736 .828 1217 H .0491 .164 .327 .491 .655 .818 .982 115 I.3I 1.47 684.4 5/16 .0767 .256 .511 .767 1.02 1.28 1.53 1.79 2.04 2.30 438.1 H .110 .368 .736 1. 10 1-47 1.84 2.21 2.58 2.94 3.3> 305.4 7/1 s .150 .501 I.OO 1.50 2.00 2.51 3.01 3-5i 4.01 4.51 224.0 >i .196 .654 1. 21 1.96 2.62 327 3.93 4.58 5-23 5.89 171.4 9/16 .24S .828 1.66 2.49 3.31 4.14 4.97 5.80 6.63 7.46 135.5 ^. .307 1.02 2.05 3-07 4.09 5.II 6.14 7.16 8.66 8.18 9.20 109.5 r .371 1.24 2.48 3.71 4.9s 6.19 7.42 9.90 II. I 90.6 •442 1.47 2.94 4.42 5.89 7.36 : 8.83 10.3 11.8 13.3 76.0 r .518 1-73 346 519 6.91 8.64 10.4 12. 1 13.8 15.6 70.5 .601 2.00 4.01 6.01 8.02 10.0 12.0 14.0 16.0 18.0 55-9 '5/16 .690 2.30 4.60 6.90 9.20 11.5 1 13.8 16. 1 18.4 20.7 48.7 I .785 2.62' ' 5.24 7.85 10.5 13." 1 15.7 18.3 20.9 23.6 • 42.8 I ^/i6 .887 2.96 5.91 8.87 11.8 14.8 , 17.7 20.7 23.6 26.6 37.9 iH •994 3-31 6.63 9-94 ^3-3 16.6 19.9 23.2 26.5 29.8 33-8 I 3/16 I. II 3.69 7.38 II. I 14.8 18.5 22.2 25.8 29.5 33-2 30.3 '^, 1-23 4.09 8.18 12.3 16.4 20.5 ' 24.5 28.6 32.7 36.8 27.3 I 5/x6 ^•35 45' 9.02 13.5 18.0 22.6 27.1 31.6 36.1. 40.6 24.9 I^ 1.48 4.95 9.90 14.9 19.8 24.8 29.7 34.6 39.6 46.6 22.7 I 7/x6 1.62 5.08 10.2 16.2 20.3 25.9 32.5 35. 5 40.6 48.7 20.7 I>i 1.77 5.89 II. 8 17.7 23.6 29.5 35.3 41.2 47-1 53-0 19.0 I 9/16 1.92 6.39 12.8 19.2 25.6 32.0 \ 38.4 44.7 51.1 57.5 '7.5 ifi 2.07 6.91 13.8 20.7 27.7 34.6, 41.5 48.4 55-3 62.9 16.2 I"A6 2.24 7.46 14.9 22.4 29.8 37.31 ^\ 52.2 59.6 67.1 15.0 I^ 2.41 8.02 16.0 24.1 32.1 40.1 1 56.1 64.1 72.2 13.9 I » 3/16 2.58 8.60 17.2 25.8 34-4 43.0 51.6 60.2 68.8 77-4 13.0 I^ 2.76 9.20 18.4 27.6 36.8 46.0 55.2 64.4 73.6 82.8 12.2 1*5/16 2.95 9.83 19.7 29.5 39.3 49.1 59.0 68.8 l^'^o 88.4 11.4 2 3.14 10.5 20.9 31.4 41.9 52.4 j 62.8 73-3 83.8 94.3 10.7 2>^ 3.55 11.8 23.6 35-5 47.3 59.1 ; 70.9 82.8 94.6 106.4 9.47 2X 3.98 133 26.5 39.8 53.0 66.3 Z2-5 92.8 106.0 "93 8.44 2^ 4.43 14.8 29.5 44.3 59.1 73.8, 88.6 103.3 1 18. 1 132.9 7-59 2;^ 4.91 16.4 32.7 49.1 65.5 81.8, 98.2 1 14. 5 130.9 147.3 6.84 2>i 5.41 18.0 36.1 54.1 72.2 90.2 108.2 126.2 144.3 162.3 6.21 2.V 5-94 19.8 39.6 59.4 79.2 99.0 118.8 138.5 158.4 178.2 5.66 2J^ 6.49 21.6 43-3 64.9 86.6 108.2 129.8 151.5 1 73. 1 194.8 5.18 3 7.07 23.6 47.1 70.7 94.3 117.8 141.4 164.9 188.5 212.1 4.7s 3^ 8.30 27.7 55-3 83.0 no. 4 138.3 ' 165.9 193.6 221.2 248.9 4.05 3>^ 9.62 32.1 64.1 96.2 128.3 160.4 192.4 224.5 256.6 288.6 3-49 3^ II.O 33-5 23-^ 1 10.4 147.3 164. 1 220.9 257.7 294.5 33'.3 3.04 4 12.6 41.9 83.8 125.7 167.6 209.4 251.3 293.2 335.0 377.0 2.67 ^^ 14.2 47.3 94.6 141. 9 189. 1 236.4 283.7 331.0 378.3 425.6 2.37 4H 15.9 53.0 106.0 159.0 212.1 265.1 319. 1 371. 1 424.1 477.1 2.11 aH 17.7 59.1 1 18. 1 177.2 236.3 295.3 354.4 413.5 472.5 531.6 1.90 5 19.6 65.5 130.9 196.4 261.8 327.3 392.7 458.2 523.6 589.1 1.71 5^ 21.7 72.2 144.3 216.5 288.6 360.8 432.9 505.1 577-3 649.4 1.55 5>^ 23.8 79.2 158.4 237.6 316.7 396.0 475.2 554.3 633.6 712.7 I.41 \^ 26.0 86.6 173. 1 259.7 346.2 432.8 519.3 605.9 692.4 779.0 1.29 6 28.3 94.2 188.5 282.7 377.0 471.2 565.5 659.7 754.0 848.2 1. 19 ROUND IRON. 241 Weight of Round Iron. Length in Feet. T\m a «.* Sect. L«n^h to weigh ^"^*'- 1 Arka. I z a 3 4 s 6 7 8 9 I ton. 'inches. sq. in. 33-2 cwts. .9876 cwts. 1.975 cwts. 2.963 cwts. 3.950 cwts. 4.938 cwts. 5.926 cwts. 6.613 cwts. 7.901 cwts. 8.888 feet 20.2 6>4 7 38.5 1. 145 2.291 3.436 4.582 5727 6.872 8.018 9.163 10.31 17.5 7}i 44.2 I.315 2.629 3-944 5.258 6.573 7.887 9.202 10.52 11.84 15.2 8 503 1.496 2.992 4.448 5.984 7.480 8.976 10.47 11.97 13.46 13.4 8^ 56.7 1.689 3.378 5.067 6.756 8.444 10.13 11.82 13.50 15.20 II.8 9 63.6 1.893 3.786 5.680 7572 9.46 11.36 13-25 '5i4 17.04 10.6 9^ 70.9 2. 1 10 4.220 6.329 8.440 10.55 12.66 14.77 16.88 18.99 9.48 10 78.5 2.338 4.676 7.012 9.352 11.69 1403 16.37 18.70 21.04 8.56 10)^ 86.6 2.577 4754 Z-7?' 10.31 12.89 15.46 18.04 19.02 23-19 7-76 II 1 950 2.828 5.656 8.485 II. 31 14.14 16.97 19.80 22.62 25.46 7.07 Il>^ ; 103- 9 3.088 6.176 9.265 12.35 15-44 18.53 21.62 24.70 27.80 6.47 1 12 ! 1131 3366 6.732 10.10 13.46 16.83 20.20 23-56 26.93 30.29 5.94 I2>^ ; 122.7 3656 7.312 10.96 14.62 18.28 21.91 25-59 29.25 32.90 5.48 13 '132.7 3.950 7.900 tiM 15.80 19.75 23.70 27.65 31.60 35.15 38.34 5.06 i3>^ 143. 1 4.260 8.520 12.78 17.04 21.30 25.56 29.82 34.08 4.70 14 153.9 4.581 9.162 13.74 18.32 22.90 26.49 32.07 36.65 41.23 437 14^ 1 165.1 4.915 9.830 1474 19.66 2458 28.49 34-41 39.32 44.24 4.07 i'5 176.7 5.259 10.52 15.78 21.04 26.30 31-46 36.81 42.08 47.33 3.80 15;^ 1 188.7 5.616 11.23 16.85 22.46 28.08 32.70 39-31 41.89 44.92 50.54 3.56 16 201. 1 5.984 11.97 17.95 23.93 29.92 35.90 47.88 53.86 3.34 16X 213.8 6.364 12.73 19.09 25.46 31.82 38.18 44.55 50.92 57.28 3-14 17 227.0 6.755 13.51 20.27 27.02 33-78 40.53 47-29 54.04 60.80 2.96 ;I7^ 1 240.5 7.»59 14.32 21.48 28.64 35-5? 42.95 50.11 57.28 6443 2.79 iiS 254.5 2S3.5 Z-573 '5i5 22.72 30.29 37-86 45-44 5301 60.60 68.16 2.64 19 8.438 16.88 25.32 33.75 42.19 50.63 59.03 67.52 75.94 2.37 ao 314-2 9350 18.70 28.05 37.40 46.75 56.10 65.45 7480 84.15 2.14 21 346.4 10.31 20.62 30.93 41.23 51.54 ^J'li 72.16 82.47 92.78 1-94 22 I 380.1 II. 31 22.63 33-94 45.25 56.57 67.88 79.19 86.56 90.51 101.8 1.77 23 1 4«5-5 12.37 2473 37.10 49.46 61.83 Z^'g 93.92 III. 3 1.62 24 , 452.4 1 13-46 26.93 40.39 53.86 67.32 1 80.78 94.25 107.7 121.3 1.49 16 242 WEIGHT OF METALS. Table No. 77.— -WEIGHT OP ANGLE-IRON AND TEE-IRON. I Foot in Length. Note. — When the base or the web tapers in section, the mean thickness is to be measured. Thick- NSSS. Sum of the Width and Depth ik Inches. inches. 3/16 5/16 I^ lbs. .81 1.04 1.24 iH lbs. .62 .89 I.15 1.37 ^H lbs. .68 .97 1.25 1.50 ^H lbs. •73 1.05 1.36 1.63 lbs. .78 I.I3 1.46 1.76 ^>i lbs. .83 1. 21 1.56 1.89 2H ^H lbs. lbs. .88 .94 1.29 1.37 1.67 1.77 2.02 2.15 ^'A lbs. .99 1.45 1.88 2.28 ^H lbs. 1.04 1.52 1.98 2.41 2U lbs. 1.09 1.60 2.08 2.54 3/i6 S/16 7/16 2% 1. 14 1.68 2.19 2.67 3.13 3.57 1.20 1.76 2.29 2.80 3.28 3.75 3'A aX sH zA 1.25 1.84 2.40 2.93 3-44 3.93 1.30 1.91 2.50 306 3-59 4. 1 1 1.45 1.99 2.60 3.19 3.75 4.29 1. 41 2.07 2.71 3.32 3-91 4.48 ZH 1.46 2. IS 2.81 3-45 4.06 4.66 zH 1.51 2.23 2.92 3.58 4-22 4.84 3% 1.56 2.30 3.02 371 4.38 5.02 1.62 2.38 3.13 3.84 4.53 5.20 A% 1.72 2.54 3-33 4.10 4.84 5.56 3/16 S/x6 H 7/16 91x6 4H aU 2.70 3.S4 436 5.16 5.92 6.67 7.38 2.85 3-75 4.62 5-47 6.29 7.08 7.85 3.01 4.88 5.78 6.65 7.50 8.32 5X 3.16 4.17 5- 14 6.09 7.02 .92 .79 I sH 5H 6X 6>i 3.32 4.38 5.40 6.41 7.38 8.33 9.26 3.48 4.58 5.66 6.72 7.75 8.75 9.73 3.63 4-79 5.92 7.03 8.11 9.17 10.20 3 79 5.00 6.18 7-34 8.48 9.58 10.66 3.95 5.21 6.45 7.66 8.84 10.00 II. 13 6^ 4.10 5.42 6.71 7.97 9.21 10.42 11.60 4.26 5-63 6.97 8.28 10.83 12.07 5/16 H 7/16 9/x6 H 1% 5-83 7.23 8.59 9.93 11.25 12.54 i^8o rA 7^ 6.04 6.25 7-49 7.75 8.91 9.22 10.30 10.66 11.67 12.08 1301 13.48 14.32 14.84 8 8X 8K 6.46 8.01 9.53 11.03 12.50 13.94 15.36 6.67 8.27 9.84 11.39 12.92 14.41 15.89 6.88 8.53 10.16 11.76 '3-33 15.88 16.41 8^ 9X ^% 7.08 8.79 10.47 12.12 13.75 15.35 16.93 7.39 9.05 10.78 12.49 14.17 15.82 17.45 750 9.31 11.09 12.85 14.58 16.29 17.97 7.71 9.57 II. 41 13.22 15.00 16.76 18.49 9^ 7.92 9.83 11.72 13.58 15.42 17.23 19.01 7/16 'A 91x6 zo loK 12.03 12.66 13.95 14.67 15.83 16.67 17.70 18.63 19.53 20.57 23.13 24.38 ZI ">i 12 13.28 15.40 17.50 19.57 21.61 25.63 13.91 16.13 18.33 20.51 22.66 26.88 14.53 16.86 19.17 21.44 23.70 28.13 ia>^ 17.59 20.00 22.38 24.74 29.37 13 13K 14 14^ 18.31 20.84 23.31 25.78 30.63 19.04 21.67 24.25 26.83 31.88 19.77 22.50 25.19 27.87 33.13 20.50 2334 26.12 28.91 34.38 15 21.22 24.17 27.06 29.95 3563 12 25.70 28.13 32.45 36.67 izV 24.74 29.37 33.91 38.33 13 13K 25.78 30.63 35.36 40.00 26.83 31.88 36.82 41.67 14 27.87 38.28 43-33 15 29.95 3563 41.19 46.67 16 17 18 19 20 32.03 38.13 44.12 50.00 34.12 40.63 47.02 53-33 36.20 41.13 49.95 56.67 38.28 43.63 52.87 60.00 40.36 46.13 55.78 63.33 WROUGHT-IRON PLATES. 243 Table No. 78.— WEIGHT OF WROUGHT-IRON PLATES. 1 Sect. Akra, when I foot Area in Square Feet. Number Thick- mss. of sq. ft. in wide. I 2 3 4 5 6 7 8 9 X ton. inch«. sq. in. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. sq. feet. X 3.00 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 224.0 5/16 3.75 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100. 112.5 179.2 H 4.50 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0 I35-0 149.3 7/16 5.20 17.5 35.0 52.5 70.0 87.5 105.0 122.5 140.0 157.5 180.0 128.0 ^ 6.00 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 112.0 9/16 I 6.75 22.5 45.0 67. 5 90.0 112.5 , 135.0 150.0 180.0 202.5 99.67 H l^"" 25.0 50.0 75.0 lOO.O 125.0 1 150.0 175.0 200.0 225.0 89.60 «»A6 1 8.25 27.5 55.0 82.5 ZIO.O 137.5 165.0 192.5 220.0 247.5 81.45 H 9.00 30.0 60.0 90.0 120.0 150.0 180.0 210.0 240.0 270.0 74.67 »3/i6 9.75 32.5 65.0 97.5 130.0 162.5 195.0 227.5 260.0 292.5 68.92 ^ 11.50 35.0 70.0 105.0 140.0 175.0 210.0 245.0 280.0 315.0 64.00 ^s/x6 11.25 37.5 75.0 112.5 150.0 187.5 ' 225.0 262.5 300.0 337.5 59-73 X- 12.00 40.0 80.0 120.0 160.0 200.0 240.0 280.0 320.0 360.0 56.00 I V«6 12.75 42.5 85.0 127.5 170.0 212.5 255.0 297.5 3+0.0 382.5 52.71 I^ 13.50 450 90.0 1350 z8o.o 225.0 270.0 3150 360.0 405.0 49.78 I 3/z6 14,25 47.5 95.0 142.5 190.0 237.5 285.0 332.5 380.0 427.5 47.16 x>< 15.0 50.0 lOO.O 150.0 200.0 250.0 300.0 350.0 400.0 450.0 44.80 I^ 16.5 55-0 1 10.0 165.0 220.0 275.0 1 330.0 385.0 440.0 4950 40.73 I^ 1 18.0 60.0 120.0 iSo.o 240.0 300.0 360.0 420.0 480.0 540.0 37.33 l|F 21.0 1 70.0 140.0 210.0 280.0 350.0 420.0 490.0 560.0 630.0 32.00 2 24.0 80.0 160.0 240.0 320.0 400.0 480.0 560.0 640.0 720.0 28.00 cwts. cwls. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 2}4 30 .893 1.79 2.68 3-57 4.46, §■36 6.25 Z-'^ 8.04 25.40 3 36 1.07 2.14 3.21 4.29 5.36 6.64 7.50 8.57 9.64 18.67 3->^ ^ 1.25 2.50 3.75 5.00 6.25 ^50 s.57 8-75 10.00 11.25 16.00 4 48 1.43 2.86 4.29 571 7.14 10.00 "•53 12.86 14.00 4>i 54 1. 61 3.21 4.82 6.43 8.04 9.64 11.25 12.86 14.46 12.44 5 60 1.79 3.57 5.36 7.14 8.93 10.71 12.50 14.29 16.07 11.20 5>^ 66 1.96 3.93 5.S9 7.86 9.82 11.79 13.75 15.71 17.68 10.18 6 72 2.14 4.29 6.43 8.57 10.71 12.86 15.00 17.14 19.29 9-33 7 84 2.50 5.00 Z'5° 10.00 12.50 1500 17.50 20.00 22.50 8.00 8 95 2.86 5.71 8.57 "43 10.29 17.14 20.00 22.86 25.71 7.00 9 108 3-21 6.43 9.64 12.86 16.07 19.29 22.50 25.71 28.93 6.22 10 120 3.57 7.14 10.71 14.29 12.86 21.43 25.00 28.56 32.14 5.60 Ji 132 3-93 7.86 11.79 15.71 19.64 2357 27.50 31.43 35.36 5.09 12 144 4.29 8.57 Z2.86 17.14 18.57 21.43 25.71 30.00 34.29 38.57 4.67 J3 '5^ 4.64 9.29 13.93 23.21 27.86 32.50 37.14 41.79 431 14 168 5.00 10.00 15.00 20.00 25.00 1 30.00 3500 40.00 45.00 48.21 4.00 •' il 180 5.36 10.71 16.07 21.43 26.79 1 32.14 37.50 42.86 3-73 244 WEIGHT OF METAI^S. Table No. 79.— WEIGHT OF SHEET IRON. AT 480 LBS. PER CUBIC FOOT. According to Wire-gauge used in South Staffordshire (Table No. 17). Thickness. Area in Square Fbrt. B.W.G. 32 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 IS 14 13 12 II 10 9 8 7 6 5 4 3 2 I inch. .0125 .0141 .0156 .0172 .0188 .0203 .0219 .0234 .0250 .0281 .0313 •0344 .0375 .0438 .0500 •0563 .0625 .0750 .0875 .1000 .1125 .1250 .1406 .1563 .1719 .1875 .2031 .2188 .2344 .2^00 .2813 .3125 lbs. .500 .562 .625 .688 .750 .813 .875 .938 1. 00 113 I.2q 1.38 I. SO I.7S 2.00 2.25 2.50 3-00 3- SO 4.00 4.50 5.00 5.63 6.25 6.88 7.50 813 8.75 9.38 10.0 11.25 12.5 lbs. 1. 00 I.I3 1.25 1.38 1.50 1.63 1.88 2.00 2.25 2.50 2.75 300 3. SO 4.00 4.50 5.00 6.00 7.00 8.00 9.00 lO.O "3 12.5 13.8 15.0 16.3 17.S 18.8 20.0 22.5 25.0 lbs. 1.50 1.69 1.88 2.06 2.25 2.44 2.63 2.81 3.00 3.38 3.7s 4.13 4.50 5.25 6.00 6.75 7. SO 9.00 10.5 12.0 13.5. 15.0 16.9 16.8 20.6 22.5 24.4 26.3 28.1 30.0 33.8 37.5 lbs. 2.00 2.25 2.50 2.75 3.00 3.2s 3.50 3.7s 4.00 4.50 5.00 5.50 6.00 7.00 8.00 9.00 10.0 12.0 14.0 16.0 18.0 20.0 22.5 25.0 27-5 30.0 32. S 3S.O 37. S 40.0 45.0 50.0 lbs. 2.^0 2.81 313 3-44 3-75 4.06 4.38 4.69 S-oo S.63 6.25 6.88 7.50 8.75 10. o "•3 12.5 150 17.5 20.0 22.5 2q.O 25.1 31-3 34.4 37.S 40.6 43.8 46.9 50.0 56.3 62.S lbs. 3.00 3.38 3.75 4.13 4- JO 4.88 5.2s S.63 6.00 6.75 7.50 8.25 9.00 10.5 12.0 13.5 IJ.O 18.0 21.0 24.0 27.0 30.0 33.8 37. S 41.3 45.0 48.8 S2.S 56.3 60.0 67.5 75.0 lbs. 3-50 3.94 4.38 4.81 5-2S 5.69 6.13 6.56 7.00 7.88 8.7s 9.63 10.5 12.3 14.0 15.8 I7.S 21.0 24. S 28.0 31.5 3S.O 49-4 43.8 48.1 S2.5 56.9 61.3 65.6 70.0 78.8 87. S 8 lbs. 4.00 4.50 5.00 S-So 6.00 6.50 7.00 7.50 8.00 9.00 lO.O II.O 12.0 14.0 16.0 18.0 20.0 24.0 28.0 32.0 36.0 40.0 45.0 50.0 SS.o 60.0 65.0 70.0 7S.0 80.0 90.0 100.0 lbs. 4.50 5.06 5.63 6.19 6.75 7.31 7.88 8.44 9.00 10. 1 "3 12.4 13.S 18.0 20.3 22.5 27.0 31.S 36.0 40.5 45.0 50.6 S6.3 61.9 67.5 72.1 78.8 84.4 90.0 101.3 112.5 Number of sq. ft. in I ton. sq. ft 4480 3986 3584 32S6 2987 275s 2560 2388 2240 1982 1792 1623 1493 1280 1 120 996 896 747 640 560 498 448 398 358 326 299 276 256 239 224 199 179 IRON SHEETS. 24s Table No. 80.— WEIGHT OF BLACK AND GALVANIZED IRON SHEETS. (Morton's Table, founded upon Sir Joseph Whitworth & Co.*s Standard Birmingham Wire-Gauge.) XoTE. — ^The numbers on Holtzapflfel's wire-gauge are applied to the thicknesses on Whitworth's gauge. Gauge of Black Sheets. Approximate number of sqtiare feet in i ton. Gauge of Black Sheets. Approximate ntmiber of square feet in x ton. Wire- Thickness. Black Galvanized Wire- Thickness. Black Galvanized Gauge. Sheets. Sheets. Gauge. Sheets. Sheets. No. inch. square feet square feet. No. inch. square feet square feet I .300 187 185 17 .060 933 876 2 .280 200 197 18 .050 1120 1038 3 .260 215 212 19 .040 1400 1274 4 .240 233 229 20 .036 1556 1403 5 .220 ^§^ 250 21 .032 1750 1558 6 .200 2^^ 275 22 .028 2000 1753 7 .180 3" 304 23 .024 2333 2004 8 .165 339 331 24 .022 2545 2159 9 .150 373 363 ^§ .020 2800 2339 10 .135 415 403 26 .0x8 3111 ?^ II .120 467 452 H .016 3500 12 .110 509 491 28 .014 4000 3122 '3 .095 589 566 29 .013 4308 3306 14 .085 659 630 30 .012 4667 3513 15 .070 800 757 31 .010 5600 4017 16 1 .065 862 813 32 .009 6222 4327 246 WEIGHT OF METALS. Table No. 81.— WEIGHT OF HOOP IRON. I FOOT IN LENGTH. According to Wire-gauge used in South Staffordshire. Thicj Width in Inches. CNBSS. H ^ H I ^}i IX iH i>4 B.W.G. inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 1 lbs. 21 20 19 ■0344 .0375 .0438 .0716 .0781 .0911 .0861 .0938 .109 .100 .109 .128 .115 .146 .129 .141 .164 .144 .156 .182 .158 .172 .200 .172 .188 .219 18 16 .0500 .0563 .0625 .104 .117 .130 .125 .141 .156 .146 .164 .182 .167 .188 .208 .188 .211 .234 .208 .234 .260 .229 .258 .286 .250 .281 .313 15 13 .0750 .0875 .1000 .156 !208 .188 .219 .250 .219 .256 .292 .250 .293 .333 .281 .329 .375 •313 .366 .416 .344 .402 •458 .375 .438 .500 12 II 10 .1125 .1250 .I40& .234 .260 .293 .281 .313 .352 .328 •365 .410 .375 .417 .469 .422 .469 .527 .469 .521 .586 .516 •573 .645 .563 .625 .703 7 .1563 .1719 .1875 .326 .358 .391 .391 .430 .469 .456 .501 .547 .522 .573 .625 •587 .644 •703 .652 .716 .781 .717 .788 .859 •783 .859 .938 6 5 4 .2031 .2188 .234+ ■423 .488 .508 :r8^ .030 • .683 .677 .729 .781 .762 .820 .879 .836 .912 .977 .931 10.0 10.7 1.02 1.09 I.I7 KNBSS. Width in Inches. IHIC ^H Il< I^ a 2X 2>i 2J< 3 B.W.G. inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 21 20 19 .0344 .0375 .0438 .197 .203 .238 .201 .219 .257 .215 .224 .274 .229 .250 .292 .258 .251 .328 .287 .313 .365 •315 •344 .400 .344 •375 .437 18 17 16 .0500 .0563 .0625 .271 .305 •339 .292 .328 .365 .312 .351 .391 •333 •375 .417 .375 .422 .469 .417 .469 .521 .458 .516 .573 .500 .563 .625 15 13 .0750 .0875 .1000 .307 .475 •543 .438 .584 .469 •549 .626 .585 .667 .562 .658 .750 .625 .833 .687 .804 .917 .750 .875 I.OO 12 II 10 .1125 .1250 .1406 .609 .677 .762 .656 .729 .820 .703 .781 .879 .750 •833 .938 .842 .937 1.06 .938 1.04 1.17 1.03 1. 15 1.29 I.I3 1.16 7 .1563 .1719 .1875 .848 .931 1.02 •913 1. 00 1.09 .978 1.07 1.17 1.04 i.iS 1.25 I.I7 1.29 1.41 1.30 1^43 1.56 1.58 1.72 1.56 1.72 1.88 6 5 4 .2031 .2188 .2344 1. 10 1.19 1.27 1.28 1.37 1.27 1.37 1.46 1-35 1.46 1.56 1.64 1.76 1.69 1.82 i'9S 1.86 2.C0 2.15 2.03 2.19 2.35 WARRINGTON IRON WIRE. 247 Table No. 82.— WEIGHT AND STRENGTH OF WARRINGTON IRON WIRE. Table of Wire manufactured by Rylands Brothers. Note.— The Wire-Gauge is that of Rylands Brothers. 1 i 1 Specific 1 Size on ; Weight of II Length of j Breaking Strain. | Density, Wire- 1 Gauge. Diameter. 100 Yds. iMUe. T Bundle of 63 lbs. I Cwt. 1 An- nealed. Bright. the aver- age den- sity of iron =1. inch. milli- metres. lbs. lbs. yards. yards. , lbs. lbs. average iron = I. 7/d H 12.7 193.4 3404 33 il 10470 15700 I.OI68 «/o ^Vsa II.9 170.0 2991 37 66 9200 I381O 5/0 7/16 II.I 148. 1 2606 43 It 8020 12000 4/0 '3/3a 10.3 127.6 2247 49 88 1 6910 10370 3/0 H n 108.8 I915 r 103 5890 8835 I.O168 Vo "/3a 91.4 1609 69 123 4960 7420 .326 8.3 82.1 1447 77 136 4450 6678 I .3a> 7.6 69.6 1227 ^ 161 3770 5655 2 .274 7.0 5?-^ 1022 108 ' 193 3140 4717 3 .250 (1) 6.4 48.4 851 130 232 2618 3927 I.OI68 4 .229 5.8 40.6 714 *55 276 1 2197 3295 5 .209 5.3 33.8 595 186 332 1830 2740 6 .191 4.9 28.2 495 223 397 1528 2290 7 .174 4.4 23.4 412 269 479 1268 1900 8 •159 4.0 19.6 344 322 573 1060 1558 9 .146 3.7 16.5 290 3^ 680 893 1340 10 •133 3-4 13.7 241 460 819 741 I no 10^ .125 (i) 3.2 12. 1 213 521 927 654 980 I.OI79 II .117 3-0 10.6 186 595 1059 573 860 12 .100 (A) 2.6 8.0 142 783 1393 436 650 13 .090 2.3 n no 1006 1790 339 509 14 .079 2.0 !5 1305 2322 261 390 IS .069 1.8 3.7 65 1715 2188 3052 199 299 16 .0625(A) 1.5 2.9 51 3894 156 233 1.0690 17 •053 1-3 2.2 38 2900 5160 118 176 18 .047 1.2 1.7 30 3687 6560 93 138 19 .041 I.O 1.3 23 4847 8620 70 ^25 20 .036 .9 1.0 18 5985 11120 54 81 21 .03125(A) .8 .8 H 7574 14152 43 64 1. 1765 22 .028 .7 .6 II 9893 18486 33 49 A/^m. This Table of the weight and strength of Warrington wire is given by permission of Messrs. Rylands Brothexs; and it is said to be based on very accurate measurements of sires and weights. The last column is added by the author, to show that the density of the wire is stationary for diameters of from yi inch to X inch, and probably somewhat smaller diameters; but that, contrary to current opinions of the density of wire, the density becomes greater when the diameter is reduced to }i inch, and is gradually increased as the diameter is further reduced. 248 WEIGHT OF METALS. Table No. 83.— WEIGHT OF WROUGHT-IRON TUBES, Bv Internal Diameter. Length, i Foot, Thickness by Holuapffel's Wire-GauBe- WROUGHT-IRON TUBES. 249 Table No. 83 (continued). Length, i Foot. Thickness by Holtzapffers Wire-Gauge. Thick- ) MCSS. W. G. 8 9 10 II 12 13 14 15 16 17 18 Inch. .165 .148 •134 .120 .109 .095 .083 .072 .065 .058 .049 "/64^- 9/64/ 9/64 b. %b. 7/64 3/32/ 5/64/ s/64^. V16/ Vi6^. 3/64/ Int. DiAM. lbs. lbs. lbs. lbs. lbs. 1 lbs. lbs. lbs. lbs. lbs. lbs. inches. H .501 .423 .364 •3*8 .267 .219 .181 .149 .130 .Ill .0895 % .717 .610 •539 .472 .410 •343 .290 .243 .215 .187 •»54 H .934 .797 .714 .625 .553 .468 .398 .337 .300 .263 .218 % 1.15 1. 00 .890 ■779 .695 .592 .507 •431 .385 .339 .282 H 1.58 1-39 1.24 1.09 .981 .841 .718 .620 •555 .491 .410 1 2.01 1.78 1-59 1.41 1.27 1.09 .935 .808 •Z^5 .643 .538 iH 2.45 2.17 1.94 1.72 1.55 1.34 I.I5 .997 .895 .795 .667 ^% 2.88 2.55 2.29 2.04 1.84 1.59 1-37 1. 19 1.07 .946 .795 ^M 3-3' 2.94 2.64 2.35 2.12 1.84 '•59 1.37 1.24 1. 10 .923 2 3.74 3-33 3.00 2.66 2.41 2.08 1. 81 1.56 1.41 1.25 1.05 2% 4.17 3.72 3-35 2.98 2.69 2.33 2.02 1.75 1.58 1.40 1. 18 2% 4.61 4.10 3.70 329 2.98 2.58 2.24 1.94 1.75 1.55 1.31 2^ 504 ^2S 4.05 3.61 3.26 2.83 2.46 2.13 1.92 Hi 1.44 3 ^ 5.47 4.88 4.40 392 3.55 3.08 2.68 2.31 2.09 1.86 '•57 Z}i 6.33 5.65 5.10 ^•55 4.12 3.58 3" 2.69 2.43 2.16 1.82 4 7.20 6.43 5.80 5.18 4.69 4.07 3.55 307 2.77 2.47 2.08 4K 8.06 7.20 6.50 5.81 5.26 4.57 3.98 3-45 3." 2.77 '2.34 5 8.93 7.98 7.21 6.44 5.83 5.07 4.42 3.83 3.45 3.07 2.59 \^ 9.79 8.75 7.91 7.06 6.40 5.57 4.85 4.20 3.79 3|! 2.85 6 10.7 9.53 8.61 7.69 6.97 6.07 5.29 4.58 4. 13 3.68 3. 1 1 1 6j^ 11.5 10.3 9.31 8.32 7.55 6.56 §•72 4.96 4.47 3.98 3.36 ' 7 12.4 ii.i 10. 8.95 8.12 7.06 6.16 5-33 4.81 4.29 3.62 l^ 133 II. 9 10.7 958 8.69 Z-55 6.59 5.71 5.15 4.59 3.88 8 14. 1 12.6 11.4 10.2 9.26 8.06 7.03 6.09 5-49 4.90 4.13 9 15.8 14.2 12.8 II. 5 10.4 9.05 7.90 6.84 6.17 5.50 4.65 10 17.6 »5.7 14.2 12.7 II.5 lO.O 8.77 7.60 6.85 6.11 5.16 II 19.3 ^7-3 15.6 14.0 12.7 II. 964 8.35 7-53 6.72 5.67 12 21.0 18.8 17.0 15.2 13.8 12.0 10.5 9.10 8.21 7.33 6.19 13 22,7 20.4 18.4 16.5 15.0 130 11.4 9.86 8.89 7-93 6.70 14 245 21.9 19.8 17.7 16. 1 14.0 12.2 10.6 9.57 8.54 7.22 '5 26.2 23-5 21.3 19.0 17.2 15.0 13.1 11.4 10.3 Kl 7.73 16 1 27.9 25.0 22.7 20.3 18.4 16.0 14.0 12. 1 10.9 9.88 8.24 I '7 29.6 26.6 24.1 21.5 19.5 17.0 14.9 12.9 11.6 10.4 8.76 ' 18 * 3»-4 28,1 255 22.8 20.6 ; 18.0 15.7 13.6 12.3 II. 9.27 ;i9 33.1 29.7 26.9 24.0 21.8 1 19.0 16.6 14.4 130 11.6 9.78 • 20 34-8 31.2 28.3 25-3 22.9 20.0 ^Z-5 15.1 13.7 12.2 10.3 10.8 21 36.6 32.8 29.7 26.? 24.1 21.0 18.3 ^H 14.3 12.8 22 38.3 34.3 311 27.8 25.2 1 22.0 19.2 16.6 15.0 13.4 II. 3 23 40.0 35.9 32.5 29.1 26.4 1 23.0 20.1 17.4 15.7 14.0 II. 8 24 41.8 37.4 33-9 303 27.5 1 24.0 20.9 18. 1 16.4 14.6 12.6 26 45.2 48.7 40.5 367 32.8 29.8 1 26.0 22.6 197 17.7 15.8 134 28 43-6 39.5 35.3 32.1 28.0 24.4 21.2 19. 1 17.0 14.4 30 52.1 46.7 42.3 37.8 34.4 30.0 26.1 22.7 20.5 18.3 15.4 32 55.5 49.8 48.0 40.4 36.7 32.0 27.9 24.2 21.8 19.5 16.5 34 59.0 52.9 42.9 39.0 34.0 29,7 25.8 23.2 20.7 ^I'i >36 62.4 56.0 50.8 45-4 41.3 1 36.0 31-4 27.3 24.6 21.9 18.6 250 WEIGHT OF METALS. Table No. 84.— WEIGHT OF WROUGHT-IRON TUBES, BY External Diameter. Length, i Foot. Thickness by Holtzapffel's Wire- Gauge. Thickness. W. G. 7 8 9 ID II la 13 14 15 Inch. .180 .165 .148 .134 .120 .109 .095 .083 .072 3/x6 ^. "/64 ^• 9/64/ 9/64 ^. yib- 7/64 3/33/ 5/64/ 5/64 b. Ext. Diam. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. I inch. '•55 1.44 1-32 1.22 I. II 1.02 .900 .797 .700 'H 1.78 1.66 I.5I 1-39 1.26 1. 16 1.03 .906 •794 '^ 2.02 1.88 I.7I I.S7 1.42 1.30 1. 15 1 I.OI .888 'H 2.25 2.09 1.90 1.74 1.58 1.45 1.27 ! 1. 12 .983 '^ 2.49 2.31 2.10 1.92 ^'P 1.59 1.40 ! 1.23 1.08 iH 2.72 2.52 2.29 2.09 1.89 H'^ 1.52 1 1.34 1.17 'H 2.96 2.74 2.48 2.27 2.05 1.87 1.65 , 1.45 1.27 m 3.19 2.96 2.68 2.45 2.21 2.02 1.77 I 1.56 1.36 2 3-43 3.17 2.87 2.62 2.36 2.16 1.90 1 1.67 1-45 i% 3.67 3-39 3.06 2.80 2.52 2.30 2.02 1.78 1-55 ^% 3.90 3.60 326 2.97 2.68 2.44 2.14 1.88 1.64 2H 4.14 3.82 3.45 3.15 2.83 2.59 2.27 1.99 1.74 *^ 4.37 4.04 3.65 332 2.99 2.73 2.39 2.10 1.83 2H 4.61 4.25 3.84 350 3.15 2.87 2.52 2.21 1-93 2¥ 4.84 4.47 4.03 3.67 3-31 3.02 2.64 2.32 2.02 2^ 5.08 4.68 4.23 3.85 3-46 3.16 2.77 2.43 2.11 3 , 5.32 4.90 4.42 4.02 3.62 3.30 2.89 2.54 2.21 yx VI 5.33 4.81 4-37 3-94 3-59 314 2.75 2.40 i'A 6.26 5.76 5.20 4.72 4.25 387 3.39 2.97 2.59 2K 6.73 6.19 5.58 5-07 4.88 4.16 3.64 3.19 2.77 4 7.20 6.63 5.97 5.43 4.44 3.89 340 2.96 A% 7.67 7.06 6.36 5.78 5.20 4.73 4.38 3.62 3.15 *>i 8.14 7-49 7.45 6.13 5-5' 5.01 3.84 3-34 4^ 8.61 7.91 713 6.48 5.82 5-30 4.63 4.06 3-53 s , 9.08 8.35 7.52 6.83 6.13 5.58 4.88 4.27 372 5^ 9.56 8.79 7.91 8.30 7.18 6.44 5-»7 5.13 4.49 3-90 s'4 10.0 9.22 7.88 6.76 6.15 5-38 4.71 4.09 k^ 10. 5 9.65 8.68 7.07 6.44 5.63 4.93 4.28 6 II.O ZO.I 9.07 8.23 7.39 6.73 5.87 5.14 5.36 4.47 6H 11.4 10.5 9.46 8.58 7.70 8.02 7.01 6.12 4.66 (,'A 11.9 10.9 9.85 8.93 7.30 6.37 5-58 4.8s (>K 12.4 11.4 10.2 9.28 8.33 7.58 6.62 5-79 503 7 12.9 1 1.8 10.6 9.63 8.64 7.87 6.87 6.01 5.22 1% '3-3 12.2 II.O 9.99 8.96 8.15 7.12 6.23 541 1% 13.8 12.7 11.4 10.3 9.27 8.44 7.37 6.45 5.60 l^ 14.3 131 II. 8 10.7 9-59 8.72 7.62 6.66 5-79 8 14.7 13.5 12.2 II.O 9.90 9.01 7.86 6.88 5.98 Thicknkss. W. G. 4 5 6 7 8 9 Inch. •3125 .281 .238 .220 .2 J03 * 180 .165 .148 5/16 9/3a ^5/64/ 7/33 r X3 '/64 ^ Ueb. "/64 ^- 9/64/ Ext. Diam. lbs. lbs. lbs. lbs. "7 bs. 1 lbs. lbs. lbs. 7 inch. 21.9 19.8 16.9 15.6 li ^5 2.9 II. 8 10.6 l^ 23-5 21.3 18. 1 16.8 II >•§ 3.8 12.7 II.4 8 25.2 22.7 19.3 17.9 i( 3.6 4.7 '35 12.2 8^ 26.8 24.2 20.6 19. 1 i: r.6 5-7 14.4 12.9 9 28.4 25-7 21.8 20.2 i\ H 6.6 15.3 137 9>^ 30.1 27.1 ! 23.1 21.4 i< ?.8 7.6 16. 1 14.5 10 31.7 28.6 ! 24.3 22.5 1 ^ >.8 8.5 17.0 15.3 LIST OF TABLES OF CAST IRON, STEEL, ETC. 2$ I LIST OF TABLES OF THE WEIGHT OF CAST IRON, STEEL, COPPER, BRASS, TIN, LEAD, AND ZINC. The following Tables are devoted to the specialities of manufacture in Cast Iron, Steel, and other metals, embracing the utmost range of dimen- sions to which objects in the several metals are executed in the ordinary course of practice. Thus, whilst it is customary for certain classes of Cylinders in Cast Iron — steam cylinders, for example — to be constructed according to given internal diameters, other classes are constructed according to diameters given externally, as the iron piers of railway bridges. Two distinct tables accord- ingly have been composed, showing the weights of Cylinders of various thicknesses, and of diameters as measured internally and externally. The weights of Copper Pipes and Cylinders are only calculated for in- ternal diameters, as it is not the practice to construct them to given external diameters. Brass Tubes, on the contrary, are calculated only for external diameters, as they are not ordinarily made to given internal diameters. Table No. 85. — ^Weight of Cast-iron Cylinders, i foot in length, advanc- ing, by internal measurement, from i inch to 10 feet in diameter, and from }^ inch to 2j^ inches in thickness. Table No. 86. — ^Weight of Cast-iron Cylinders, i foot in length, advanc- ing, by external measurement, from 3 inches to 20 feet in diameter, and from ^Ixt inch to 4 inches in thickness. Table No. 87. — Volume and weight of Cast-iron Balls, when the diameter is given; from i inch to 32 inches in diameter, with multipliers for other metals. Table No. 88. — Diameter of Cast-iron Balls, when the weight is given ; from yi pound to 40 cwts. Table No. 89. — ^Weight of Flat Bar Steel, i foot in length ; from ^ inch to I inch thick, and from }i inch to 8 inches in width. Table No. 90. — Weight of Square Steel, i foot in length ; from }i inch to 6 inches square. Table No. 91. — ^Weight of Round Steel, 1 foot in length; from }i inch to 24 inches in diameter. Table No. 92. — ^Weight of Chisel Steel: hexagonal and octagonal, i foot in length; from ^ inch to i^ inches diameter across the sides. Oval-flat, from J^ x ^ inch to ij5^ x ^ inch. Table No. 93. — ^Weight of one square foot of Sheet Copper; from No. i to No. 30 wire-gauge, as employed by Williams, Foster, & Co. Table No. 94. — Weight of Copper Pipes and Cylinders, i foot in length, advancing, l)y internal measurement, from }i inch to 36 inches in diameter, and from No. 0000 to No. 20 wire-gauge in thickness. 252 WEIGHT OF METALS. Table No. 95. — ^Weight of Brass Tubes, i foot in length, advancing, by external measurement, from }i inch to 6 inches in diameter, and from No. 3 to No. 25 wire-gauge in thickness. Table No. 96. — ^Weight of one square foot of Sheet Brass; from No. 3 to No. 25 wire-gauge in thickness. Table No. 97. — Size and weight of Tin Plates. Table No. 98. — ^Weight of Tin Pipes, as manufactured. Table No. 99. — Weight of Lead Pipes, as manufactured. Table No. 100. — Dimensions and weight of Sheet Zinc. (VicUe-Mon- tagne,) CAST-IRON CYLINDERS. 253 Table No. 85. — Weight of CastJron Cylinders. By Internal Diameter, i Foot Long. Int. Thickness in Inches. DiAM. X 5/16 H 7/16 >i 9/16 H "/I6 H H I inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. I 307 4.03 5.06 6.17 7.36 8.63 9.97 II.4 12.9 16. 1 19.6 ^'A 4.30 5.56 6.90 8.32 9.82 II.4 13. 1 14.8 16.6 20.4 245 2 5.52 7.09 8.74 10.5 12.3 14.2 16. 1 18. 1 20.3 247 29.5 2>i 6.75 8.63 10.6 12.6 147 16.9 19.2 21.5 23-9 29.0 344 3 , 7.9« 10.2 12.4 14.8 17.2 19.7 22.2 24.9 27.6 33.3 39.3 3>i 9.20 II.7 143 16.9 19.6 22.4 25.3 28.3 3'.3 37.6 44.2 4 ^ 10.4 13.2 16.1 19. 1 22.1 2Q.2 28.0 28.4 31.6 350 41.9 49.1 4^ II.7 14.8 18.0 22.1 245 31.5 35-0 38.7 46.2 54.0 5 , 12.9 16.5 19.8 23.4 27.0 307 345 38.4 42.3 50.5 1^-2 5^ 14-1 17.8 21.6 255 29.5 33.5 37.6 41.8 46.0 54.8 ti'^ ^ 15.3 19.4 235 27.7 32.0 36.2 40.7 45.1 49.7 59.1 68.7 Thickness in Inches. 1 H 7/x6 H 9/x6 H "/»6 H H I IH iH inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lU. lbs. 6 23.5 27.7 32-0 36.2 40.7 45.1 49.7 59-1 68.7 78.7 89.0 ^H 25.3 29.8 344 390 ^3-7 48.5 53.4 63.4 73.6 84.2 95-1 7 , 27.2 32.0 36.8 41.8 46.8 51.9 57-; 67.7 89.7 101.2 7}i 29.0 34.1 39.3 44.5 49.9 55.3 60.8 71.9 95.3 107.4 8 30.8 36.3 41.7 47.3 52.9 58.6 64.4 76.2 80.5 100.8 "35 8)^ 32.7 38.4 44.2 50.0 55.9 62.0 68.1 93-3 106.3 119. 7 ' ^. 34.5 40.5 46.6 52.8 59.0 U.t 71.8 84.8 98.2 III.8 125.8 . 9>^ 36.4 42.7 49.1 55.6 58.3 61. 1 62.0 75.5 89.1 lOJI 108.0 112.9 117.4 131.9 • 10 1 «o>^ 3».2 40.0 44.8 47.0 51.5 54.0 65.1 68.2 72.1 75-5 82.8 93.4 97.7 122.9 128.4 138. 1 144.2 ' 11 41.9 49.1 56.5 63.9 71.2 78.9 86.5 102.0 1 17.8 133.9 150.3 11;^ 43-7 51.3 58.9 66.6 745 82.3 90.2 106.3 122.7 139.4 156.5 12 45.6 53.4 61.4 69.4 83.6 85.6 939 no. 6 127.6 145.0 162.6 1 13 49.2 57.7 66.3 749 80.4 92.4 101.2 1 19.2 137.5 156.0 1749 187.2 . ^^ 52.9 62.0 71.2 89.7 99.1 108.6 127.8 147.3 167. 1 178.1 1 15 ^1 66.3 76.1 85.9 95-9 105.9 116.0 136.4 157. 1 199.4 16 70.6 81.0 91.5 102.0 112.6 123.3 145.0 166.9 189. 1 211.7 17 64.0 749 85.9 97.0 108.2 1 19. 4 130.7 153-6 176.7 186.5 200.2 224.0 18 67.7 79.2 |9o.8 102.5 "43 126. 1 138. 1 162.2 211. 2 236.2 Thickness in Inches. H 7/16 H H H H I iH iX m i^ inches. cwt. cwt. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. ' 18 .604 .707 .811 1.02 1.23 1-45 1.67 1.89 2.II 2.34 2.46 2.56 19 .637 .746 :S 1.08 1.30 1.52 1.75 1.99 2.22 2.70 20 .670 .784 113 1.36 1.60 1.84 2.08 2.33 2.58 2.83 21 .703 .823 .942 1. 19 1-43 1.68 1.93 2.18 2.44 2.70 2.96 22 .736 .861 .986 1.24 1.49 1.76 2.02 2.28 2.55 2.82 309 ' 23 .769 .900 1.03 1.29 1.56 1.83 2.10 2.38 2.66 2.94 3-22 24 .802 .939 1.07 1.35 1.63 1.91 2.19 2.48 HI 3.06 3-35 25 .835 .977 1. 12 1.40 1.69 1.99 2.28 2.58 2.88 3.18 3-48 254 WEIGHT OF METALS. Table No. 85 {continued). By Internal Diameter, i Foot Long. Int. Thickness in Inches. DiAM. H 7/x6 % H H n I 1% 1% m i}i inches. CWU. cwts. cwts. cwts. t cwts. cwts. cwts. cwts. cwts. cwts. cwts. 26 .868 1.02 1. 16 1.46 1.76 2.06 2.37 2.68 2.99 330 362 27 .901 1.05 I.2I I.5I 1.82 2.14 2.45 2.77 309 3.42 HI 28 .934 1.09 1.25 1-57 1.89 2.22 2.54 2.87 3.20 3.54 3.8S 29 .967 i.i-:« 1.29 1.62 1.96 2.29 2.63 2.97 3.31 3.66 4.01 30 .998 1.17 1.34 1.68 2.02 2.37 2.72 3.07 342 3-78 4.14 32 1.06 1.25 1-43 1.79 2.15 2.52 2.89 327 364 4.02 4.41 3i 1.13 1.32 1.51 1.90 2.29 2.67 3.07 346 3.86 4.26 4.67 36 1.20 1.40 1.60 2.01 2.42 2.83 3.24 3.66 4.08 4-50 4.94 38 1.26 1.47 1.69 2.12 ^•§§ 2.98 3.42 3.86 4.30 4.75 5.20 40 1.33 1.55 1.77 2.23 2.68 314 3.59 4.05 4.52 4.99 5.47 42 1.39 1.63 1.86 2.34 2.81 3.29 3.77 4.25 4.74 5-23 5.73 45 1.49 1.75 1.99 2.50 3.01 3.52 4.03 4.55 5.07 5-5? 6.13 48 1.59 1.86 2.12 2.66 3-21 3-75 430 4.85 5.40 5.96 6.52 Thick NBSS IN Inches. H H Ji 1 >X 1% iji i>i I^ 2 2X inches. cwts. cwts* cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 48 2.66 3.21 3-75 430 4.85 5.40 5.96 6.52 7.63 8.77 9.91 SI 2.82 3.40 3.98 4.56 S'H 5-73 6.32 6.91 8.09 9.29 10.5 54 2.99 3.60 4.21 4.82 5-44 6.06 6.69 7.31 8.55 9.82 II.I 57 3.15 3.80 4-44 5.09 5.73 6.03 6.38 7.05 7.70 8.10 9.01 10.4 II.7 60 3.32 4.00 4.67 5-35 6.71 7.41 9-47 10.9 12.3 63 3.48 4.19 4.90 5.61 6.33 7.04 7.78 8.49 9-93 II.4 12.9 66 3.64 4-39 5.13 5.88 6.62 7.37 8.14 8.89 10.4 II.9 13-5 69 3.81 4.59 5.36 6.14 6.92 7.70 8.51 9.28 10.9 12.5 14.1 72 3.97 4.78 5- §9 6.40 7.21 8.03 8.87 9.67 "•? 130 14.7 75 4.14 4.98 5.82 6.66 7-5' 8.36 9.24 10. 1 11.8 13.5 15.2 78 4.30 5-^! 6.05 6.93 7.81 8.69 9.60 10.5 12.2 14.0 15.8 81 4.46 5.38 6.28 7-19, 8.10 9.02 9.97 10.9 12.7 14.6 16.4 84 4.63 5-57 6.51 7.45* 8.40 9.35 10.3 "3 13.2 I5.I 17.0 87 4.79 5.77 6.74 7.72 8.69 9.67 10.7 II. 6 136 15.6 17.6 90 4.96 5-97 6.97 7.98 S.24 8.99 10.0 II.I 12.0 14. 1 16. 1 18.2 93 5.12 6.17 7.20 9.29 10.3 1 1.4 12.4 14.5 16.7 18.8 96 5.28 6.36 HI 8.51 9.58 9.88 10.7 II. 8 12.8 15.0 17.2 19.4 99 5-45 6.56 7.66 8.77 II.O 12.2 13.2 15.5 17-7 20.0 102 5-^i 6.76 7.89 903 10.2 "3 12.5 13-6 15.9 18.2 20.6 loq 5.78 6.95 8.12 9.29 10.5 II. 7 12.9 14.0 16.4 18.8 21.2 108 5.94 7.15 8.36 9.56 10.8 12.0 13-3 14.4 16.8 '9-3 21.8 III 6.10 7.35 f-S9 9.82 ii.i 12.3 136 14.8 17.8 19.8 22.3 114 6.27 7.55 8.82 10. 1 11.4 12.6 14.0 15.2 20.3 22.9 117 6.43 7.74 9.05 10.4 11.7 13.0 14.3 K.6 iS.o 18.2 20.9 23-5 120 6.59 7-94 9.28 10.6 12.0 13.3 14.7 18.7 21.4 24.1 CAST-IRON CYLINDERS. ^SS Table No. 86. — Weight of Cast-Iron Cylinders. By External Diameter, i Foot Long. Ext. Thickness in Inches. DiAM. 1 3/16 X 5/x6 ' I H 7/16 K 9/16 H U H I inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 3 5.18 6.75 7.98 8.25, 9.65 II.O 12.3 13.5 14.6 16.6 18.3 19.6 Z}^ 6.10 9.78! 11.5 13.2 14.7 16.2 17.6 20.3 22.6 24.5 4 , 7.02 9.20 "3 13.3 15.3 17.2 19.0 20.7 23.8 24.0 26.9 29.5 4,^ 7-94 10.4 12.9 15.2 17.5 19.6 21.7 27.7 31.1 344 5 , 8.86 II.7 14.4 17.0 19.6 22.1 24.5 26.9 31.5 35.4 39.3 5^ 9.78 12.9 15.9 18.9 21.8 24.5 27.3 29.9 35-2 39.7 44.2 6 10.7 14. 1 17.5 20.7 23.9 27.0 30.0 33.0 38.9 44.0 49.1 6X II. 6 *$•? 19.0 22.5 26.0 29.5 32.8 36.1 42.6 48.3 54.0 7 12.5 16.6 20.5 24.4 28.2 31.9 35-6 39.1 46.4 52.6 58.9 l}i 13.5 17.8 22.1 26.2 30.3 34.4 38.3 42.2 5°i 56.9 63.8 68.7 8 14.4 19.0 23.6 28.1 32.5 36.8 41.1 ^§•3 53-8 61.2 8^ 153 20.3 25-1 29.9 34.6 39-3 43-8 48.3 57.5 ^J'i 73.6 9 16.2 21.5 26.7 31.8 36.8 41.7 46.6 51.4 61.3 69.8 9H lo 17.2 18.1 22.7 23.9 28. 2 29.7 33.6 35-4 38.9 41. 1 44.2 46.6 49.4 52.1 54.5 57.5 65.0 68.7 78.4 II 19.9 26.4 32.8 39.1 45-4 51.5 57.6 63.7 76.0 87.0 98.2 12 21.8 28.8 35.9 42.8 49.7 56.5 63.2 69.8 83.4 95.6 108.0 13 23.6 31.3 38.9 46.5 54.0 61.4 68.7 75.9 90.7 104.2 117.8 14 255 33*8 42.0 50.2 58.3 66.3 74.2 82.1 98.0 112.8 127.6 15 27.3 36.2 45-* 53.8 62.6 71.2 79.7 85.3 88.2 105.4 121.3 137.4 i6 29.1 38.7 48.1 57.5 66.9 76.1 94.3 112.7 129.9 147.3 17 3'° 41. 1 51.2 61.2 71.1 81.0 90.8 100.5 120.0 138.5 157.1 i8 32.8 43-6 54.3 64.9 75.4 85.9 96.3 106.6 127.4 147. 1 166.9 19 34-6 46.0 57.3 68.6 79-7 90.8 101.8 112.8 1347 155.7 176.7 . 20 1 36.5 48.5 60.4 72.3 84.0 95.7 107.3 118.9 142.0 164.3 186.5 ' 21 38.3 50.9 63.5 75.9 88.3 100.6 112.9 125.0 149.4 172.9 196.4 1 22 40.2 53-1 66.5 79.6 92.6 105.5 118.4 131.2 156.7 181. 5 206.2 23 42.0 55.8 69.6 83.3 96.9 "0.5 123.9 1373 164.0 190.1 ^^5;0 i 24 43.8 f^§ 72.7 87.0 101.2 115.4 129.4 143-4 171.4 198.7 225:8 . 25 45-7 60.8 7S-Z 90.7 105.5 109.8 120.3 135.0 149.6 178.7 207.2 235.6 26 47.5 63.2 78.8 943 125.2 140.5 '§5-2 186.1 215.8 245.4 27 49.4 ^§•7 81.9 98.0 114.1 130. 1 146.0 161.8 193.4 224.4 2553 28 51-2 68.1 85.0 101.7 118.4 135.0 151.5 168.0 200.7 233.0 265.1 29 53-0 70.6 88.0 105.4 122.7 139.9 157.0 174.1 208.1 241.6 274.9 30 54-9 730 91. 1 109. 1 127.0 144.8 162.6 180.2 215.4 250.2 284.7 31 56.7 75-5 94.2 112.8 131.3 149.7 168.1 186.4 222.7 258.8 294.5 : 32 58.6 77.9 97.2 1 16.4 135.6 1546 173.6 192.5 230.1 267.4 304.3 33 60.4 80.4 100.3 1 20. 1 1399 159.5 179.1 198.7 204.8 237.5 276.0 314.2 |34 62.2 82.8 103.4 123.8 144.2 164.5 184.7 244.8 284.6 324.0 35 64.1 IH 106.4 127.5 148.5 169.4 190.2 210.9 252.2 293.1 333.8 36 65.9 87.8 109.5 131.2 152.7 174.3 1957 217.1 259.5 301.7 343.6 38 69.6 92.7 115.6 138.5 161.3 184. 1 206.8 229.3 274.3 3189 363.2 40 73-3 97.6 121.8 145.9 169.9 193.9 217.8 241.6 289.0 336.1 382.9 42 77.0 102.5 127.9 153.3 178.5 203.7 228.8 253.9 3037 353.3 402.5 Jl 82-5 109.8 137. 1 164.3 191.2 218.5 245.4 272.3 325.8 379.1 432.0 88.0 117.2 146.3 175.4 203.8 233.2 262.0 290.7 347.9 404.8 461.4 5« 93.6 124.6 155-5 186.4 216.5 247.9 278.6 309.1 370.0 430.6 490.9 54 99.1 131.9 164,7 197.5 229.2 262.6 295.1 327.5 392.1 456.4 520. J 57 104.6 139.3 173.9 208.5 241.8 277.4 311.7 345.9 414.2 482.1 549.8 60 IIOlI 146.6 183. 1 • 219.6 254.5 292.1 328.3 364.3 436.3 507.9 579.3 256 WEIGHT OF METALS. Table No. 86 {continued), Bv External Diameter, i Foot Long. Ext. 1 Thicknrss in Inchbs. DiAM. 3/i6 % 5/16 H 7/16 >i 9/16 H ^ Ji z ft. in. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 53 1.03 1.08 1.44 1.71 2.06 2.39 2.74 3.08 3.42 4.09 4.77 5-43 56 1.50 1.80 2.16 2.50 2.87 3.22 3.58 4.29 5.00 5-70 59 '•'2 1.55 1.88 2.26 2.62 300 3-37 3.75 4-49 523 5.96 60 1. 18 1.61 1.96 2.36 2.74 3- 14 3.52 3.91 4-69 546 6.22 63 1.23 1.67 2.05 2.45 2.85 327 3.66 4.08 4.88 5.69 6.49 66 1.28 1.73 2.13 2.55 2.97 340 3.81 4.24 5.08 5.92 6.75 69 '•33 1.78 2.21 2.65 3.09 3-53 3.96 4.41 5.28 6.38 7.01 70 1.38 1.84 2.29 2.75 3.20 3.66 4.10 4.57 5-57 7.28 76 1.48 1.95 2.46 2.95 3.43 3.92 4.39 4.90 5.87 6.84 7.80 80 1.58 2.07 2.62 3.15 3.67 4.19 4.69 5.23 6.26 7.30 8.33 Thick NBSS IN Inches. ^% iX 1^8 ^% im: 2 { cwts. 2X 2>4 2^ 3 3>^ 4 inches. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. cwts. 6 .481 .520 .557 .592 .652 .701 'W. .761 6>i •530 •575 .618 .657 .729 .789 .838 .872 .906 7 •579 .630 .678 .723 .805 .876 .938 .982 1.03 1.05 VA .629 .68s .738 .789 .882 .964 1 1.04 1.09 I.I5 1. 18 8 .678 .740 .799 .855 .959 i.os 1. 14 1.20 1.27 1.32 1.38 8>^ •727 .794 .859 •^11 1.04 1. 14 1.23 I.3I 1.39 '•45 1.53 9 .777 .849 .919 .986 I. II 1.23 1-33 1.42 I.5I 1.58 1.69 1.75 9y2 .826 .904 .980 1.05 I.I9 1 M' 1.43 1.53 1.63 I.7I 1.84 1.93 10 .875 .959 1.04 1. 12 1.27 1.40 1.53 1.64 1.75 1.84 1.99 2.10 II .974 1.07 i.x6 X.25 1.42 1.58 1.73 1.86 1-99 2.10 2.30 2.46 12 1.07 I.18 1.28 1.38 1.57 1 1.75 1.92 2.08 2.23 2.37 2.61 2.81 13 1.17 1.29 1.40 1.51 1-73 1.93 2.12 2.30 2.47 2.63 2.92 3.16 H 1.27 1.40 1.52 1.64 1.88 2.10 2.32 2.52 2.71 2.89 3.22 l& 15 1.37 1.51 1.65 1.78 2.03 2.28 2.52 2.74 2.95 3.16 \^ 16 1.47 1.62 1.77 I.9I 2.19 2.45 2.71 2.96 319 3-12 4.21 17 1.57 1.73 1.89 2.04 2.34 2.63 2.91 3.18 3.44 3.68 4.14 4.56 18 1.66 1.84 2.01 2.17 2.49 2.81 3." 3.40 3.68 3-95 4.45 4.91 20 1.86 2.06 2.25 2.43 2.80 3.16 3-50 3.83 4.16 4.47 5-^ 5.61 22 2.06 2.27 2.49 2.70 3." l^ 3.90 4.27 4.64 5.00 5.68 6.32 24 2.26 2.49 2.73 2.96 3.41 4.29 4.71 5.12 5.52 6.29 7.01 27 2-55 2.82 3.09 3.35 3.87 4.38 4.88 5.37 1-^5 6.31 7.21 8.06 30 2.85 3.»S 346 3.75 4-33 4.91 5-47 6.03 6.57 7.10 8.13 9.12 33 3.14 3.48 3.82 4.14 4.79 5-44 6.06 6.68 7.29 7.89 9.05 10.2 36 3-44 3.81 4.18 4.54 5.25 5.96 6.66 7.34 8.01 8.68 9.97 II. 2 39 3-74 4.14 4.54 4.93 5.72 6.49 7.2s 8.00 8.74 9.47 10.9 12.3 42 4-03 4.47 4.90 5-33 6.18 7.01 7.84 8.66 9.46 10.3 II. 8 13.3 45 4.33 4-79 5.26 5-72 6.64 Z-5* 8.43 9.31 10.2 II. I 12.7 14.4 48 4.62 5.12 5.62 6.12 7.10 8.07 9.02 9.98 10.9 II. 8 13.7 15.4 51 4.92 5-^5 5.98 6.51 7.56 8.59 9.61 10.6 1 1.6 12.6 14.6 16.5 54 5.22 5.78 6.35 6.91 8.02 9.12 10.2 "3 J2.4 134 15.5 17.5 57 5-51 6.11 6.71 7.30 8.48 9.64 10.8 11.9 ^K 14.2 i 16.4 18.6 60 5.81 6.44 7.07 7.70 8.94 10.2 11.4 12.6 13.8 15.0 173 19.6 CAST-IRON CYLINDERS. 257 Table No. 86 {continued). By External Diameter, i Foot Long. Ext. DiAM. i%. in. 59 60 63 66 69 70 76 80 86 90 96 100 106 11 o 116 12 o 130 140 150 160 17 o 180 190 200 THICKNESS IN Inches. ^% cwts. 6.10 6.40 6.70 7.00 7.29 7.58 7.88 8.17 8.77 9.36 9-95 10.5 II. I 11.7 12.3 12.9 13-5 14. 1 15.3 16.5 17-7 18.8 20.0 21.2 22.4 23.6 iX cwts. 6.77 7.09 7.42 7-75 8.08 8.41 8.74 9.07 9.72 10.4 II. o 11.7 12.3 13.0 13-7 14-3 15.0 15.6 16.9 18.3 19.6 20.9 22.2 23.5 24.8 26.1 ^H CWIS. 7.43 7-79 8.15 8.51 8.88 9.24 9.60 9.96 10.7 1 1.4 12. 1 12.9 13.6 14.3 150 15.7 16.5 17.2 18.6 20.1 21.5 23.0 24.4 25.9 27.3 28.8 iK cwts. 8.09 8.48 8.88 9.27 9.67 0.1 0.5 0.9 1.6 2.4 3-2 4.0 4.8 5.6 6.4 7.2 7-9 8.7 20.3 21.9 235 25.0 26.6 28.2 29.8 31-4 i^ cwts. 9.40 9.86 0.3 0.8 1.2 1.7 2.2 2.6 35 4.5 5-4 6.3 7.2 8.1 9.1 20.0 20.9 21.8 237 25.5 27.3 29.2 31.0 32.9 34.7 36.5 2 ^% cwts. cwts. 10.7 12.0 II. 2 12.6 II. 8 13.2 ".3 13-8 12.8 •14.4 13.3 14.9 13.9 15.5 ;t^ 16. 1 154 ^*1W 16.5 J^S 17.5 19.7 18.6 20.8 19.6 22.0 20.7 23.2 21.7 24.4 22.8 2S.6 23.8 26.7 24.9 27.9 27.0 30.3 29.1 32.7 31.2 35.0 33.3 ?H 35.4 ;39.8 37.5 42.2 39-6 44.5 41.7 46.9 ^}i cwts. 13-3 13.9 14.6 15.2 159 16.6 17.2 17.9 21.5 23.1 24.4 25.7 27.1 28.4 29.7 31.0 33-6 36.3 38.9 41.5 44.2 46.8 49-4 52.0 »ii cwts. 14.5 15.2 15-9 cwts. 15.8 16.6 cwts. 18.3 cwts. 17 2^^S WEIGHT OF METALS. Table No. 87. — ^Volume and Weight of Cast-Iron Balls. Given the Diameter. Diameter. Contents. Weight. Diameter. Contents. Weight. Diameter. Contents. Weight. inches. cubic inches. pounds. inches. cubic inches. pounds. inches. cubic feet. cwts. I .524 .136 8 268.1 69.8 19 2.078 8.35 1/3 1.77 .460 sy2 321.5 •83.7 20 2.424 9-74 2 4.19 1.09 9 381.7 99.4 21 2.806 11.28 2y2 8.18 2.13 9}4 448.9 I16.9 22 3.227 12.97 3 3}^ 14. 1 3.68 5-85 10 523.6 136.4 23 3,688 4.188 14.82 y 22.5 inches. cubic feet cwts. 24 16.83 4 33.5 8.73 II •403 1.62 25 4-736 19.03 4>^ 47-7 12.4 12 .524 2.10 26 5-327 21.40 5 ^ 65-5 17.0 13 .666 2.68 27 5-963 23.96 sH 87.1 22.7 14 .832 3-34 28 6.651 26.72 6 113-1 295 15 1.023 4. 1 1 29 7.390 29.69 6}^ 143-8 37.5 16 1. 241 4.99 30 8.181 32.87 7 179.6 46.8 17 1.489 5-98 31 9.027 36.27 1/2 220.9 57.5 18 1.767 7.10 32 9930 39-90 Nbie. — To find the weight of balls of other metals, multiply the weight given in the table by the following multipliers : — For Wrought Iron 1.067, making about 7 per cent. more. Steel 1.088 ,, 9 „ Brass 1.12 ,, 12 ,, Gun Metal 1. 165 „ 16)^ „ Table No. 88. — Diameter of Cast-Iron Balls. Given the Weight. .Weight Diameter. Weight Diameter. Weight Diameter. Weight Diameter. pounds. inches. pounds. inches. • pounds. inches. cwis. inches. >^ 1-54 14 4.68 80 8.37 8 18.73 I 1-94 16 4.89 90 8.71 9 19.48 2 3 4 5 2.45 2.80 3.08 332 18 20 25 28 5-09 5-27 5.68 5-90 100 9.02 10 12 14 16 20.17 21.44 22.57 23.60 cwts. I inches. 9-37 10.72 6 3.53 30 6.04 2 11.80 18 24.54 7 372 40 6.64 3 13-51 20 25.42 8 3-89 50 7.16 4 14.87 25 27.38 9 4.04 56 7-43 5 16.02 30 29.10 10 4.19 60 7.60 6 17.02 35 30.64 12 4.45 70 8.01 7 17.91 40 32.03 WEIGHT OF FLAT BAR STEEL. 259 Table No. 89. — Weight of Flat Bar Steel. I Foot Long. Width in Inches. Thickness. >i H • H I iX i>^ iH mches. lb. lbs. lbs. lbs. lbs. lbs. lbs. lbs. X .425 .533 .640 .743 .850 1.06 1.28 1.49 s/i< •53i .665 .800 .929 1.06 1.33 1-59 1.86 ^ .638 .798 .960 1. 1 1 1.28 "•59 1.91 2.23 7/16 .744 .931 I.I2 1.30 1.49 1.86 2.23 2.60 }i, .850 1.06 • 1.28 1.49 1.70 2.13 ^•55 2.98 9/16 1.20 1.44 1.67 I.91 2.39 2.87 3-35 ><, ^_ 1-33 • 1.60 1.86 2.12 2.66 3- 19 3-72 T 1.76 2.04 2.34 2.92 3-5* 4.09 — 1:92 2.23 2.5s 3.19 3.83 4.46 t' — — — 2.41 2.76 3-45 4.14 4.83 — 2.60 2.98 372 4.46 5-^1 »5/i6 — — 319 3.98 4.78 5.58 I — — — — 3-40 4^25 5.10 5-95 Width ii 4 Inches. Thickmkss. Jb ^^ *^»4^ ** Awa^k^v 2 2X 2)i 2H 3 3X 3>i 4 inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. H, 1.70 I.9I ^•33 2.34 2.55 2.76 2.98 340 s/i6 2.13 2.39 2.87 2.66 2.92 3- 19 3.45 372 4.25 H 2.98 3.19 3.51 3.83 4.14 4.46 5.10 1/16 3.35 372 4.09 4.46 4.83 5.21 ^P yi 3.40 3-83 4.25 4.68 5.10 5-53 5-95 6.80 9/16 3.83 4-30 4.78 5.26 5.74 6.22 6.69 7.6s H 4.25 4.78 5-3' 5.84 6.38 6.91 7.44 8.50 T 4.68 5.26 5.84 6.43 7.01 7.60 8.18 9.35 5.10 5.74 6.38 7.01 7.65 8.20 8.93 10.2 «3/i« 5.53 6.22 6.91 7.60 8.29 8.98 9.67 II. I n 5.95 6.69 7-44 8.18 8.93 9.67 10.4 II.9 r^'^ \ It 7.17 7-97 8.77 9.56 10.4 II. 2 12.8 7.65 8.50 9.35 10.2 II. I 11.9 13-6 ^P** *>««^ ^v ^ tf <* Width n r Inches. 4>i 5 S}i 6 eyi 7 7K 8 iiicii<es. 1 Ite- lbs. lbs. lbs. lbs. lbs. lbs. lbs. 5^ 1 3.82 4.26 4.68 §•'2 5.52 5.96 6.38 6.80 V.<s 4.78 §•32 5.84 6.38 t'^ 7-44 7.97 8.50 H 5.74 6.38 7.02 7.66 8.92 8.28 8.92 9.56 10.2 7/16 6.70 7.44 8.18 9.66 10.4 II. 2 II.9 1 5^ 1 7'^ 8.50 9.36 10.2 II. I II. 9 12.8 136 9/z6 8.60 9.56 10.5 II.5 12.4 13.4 143 iS-3 ^ 9.56 10.6 n.7 12.8 13.8 14.9 15-9 17.0 i«/i6 10.5 11.7 12.9 14.0 ^H 16.4 17.5 18.7 yr 1 II.5 12.8 14.0 15-3 16.6 17.9 19. 1 20.4 >3/i6 12.4 13.8 15.2 16.6 18.0 19-3 20.7 22.2 ■*■/■ «-^ ^ 13-4 14.9 16.4 17.9 19.4 20.8 22.3 23.8 «5/x6 14.3 15.9 '2-5 19. 1 20.8 22.4 23.9 25.6 I 15.3 17.0 18.7 20.4 22.1 23.8 25.5 27.2 26o WEIGHT OF METALS. Table No. 90. — ^Weight of Square Steel. I Foot in Length. Size. inches. V16 /A- V.6 H V.6 'V.6 Weight. pounds. •053 .119 .212 •333 .478 .651 .850 1.08 1.33 1. 61 1.92 2.24 2.60 Size. inches. 'A6 >^ v.6 I/' /'4 H v.6 2 Weight, pounds. 3.06 3-40 3.83 4.30 4.79 5.31 5.86 6.43 7-03 7.65 8.30 8.98 9-79 Size. inches. 2 2}i 2^ 2^ 2^ 3 Weight. pounds. 10.4 II. 2 II. 9 12.8 13.6 15-4 17.2 19.2 21.2 23-5 25-7 28.2 30.6 Size. inches. 4 % 4K 4' 6 Weight. pounds. 35-9 41.6 47.8 54-4 61.4 68.9 76.7 85.0 93-7 102.8 112.4 122.4 Table No. 91. — Weight of Round Steel. I Foot in Length. Diameter. Weight. Diameter. Weight. Diameter. Weight. Diameter. Weight, inches. pounds. inches. pounds. inches. pounds. inches. cwts. 'A .0417 I V.6 5.18 4 42.7 12 3.433 V.6 .0939 ^H 6.01 ^y^ 48.3 I2>^ 3729 H .167 I V.6 6,52 A% 54-6 13 4.029 V.6 .260 Ifi 7^o5 4H 60.3 is'A 4.345 H •375 I 'V.6 7.61 5 66.8 14 4.682 V.6 •511 I^ 8.18 1% 73^6 I4j4 5-013 y2 .667 I 'V.6 8.77 sy 80.8 IS . 5-364 v.6 .845 I^ 9.38 SVa 88.3 ^sH 5-728 'V.6 1.04 I 'V.6 lO.O 6 96.1 • 16 i6>^ 17 6.103 6.471 6.868 7.302 1.27 ^•50 1.76 2 10.7 12.0 13.6 inches. eyi 7 8 cwts. 1.007 I 168 'V.6 2.04 2.35 2^ 16.7 I.34I 1.526 18 19 7-724 8.607 I 2.67 2fi 18.4 sy ^•723 20 9-537 I V16 3.00 2J^ 20.2 9 , ^•931 21 10.52 i}i 3^38 2^1 22.0 9y 2.152 22 11-54 I VI6 3^76 24.1 10 2.385 23 12.62 I^ 4.17 3^ 28.3 loj^ 2.629 24 13-73 I V16 4.60 3>^ 32.7 II 2.884 I?^ 5-05 3?< 34^2 ii>^ 3^i5o WEIGHT OF CHISEL STEEL. 261 Table No. 92. — Weight of Chisel Steel — Hexagonal, Octagonal, AND Oval-Flat. I Foot in Length. Dxaneter across the Sides. HbX AGONAL SbCTION, Octagonal Section. Sectional Area. Weight. Length to weigh I cwt. Sectional Area. Weight. Leng^th to weigh I cwt inches. square inches. pounds. feet square inches. pounds. feet H .1217 .2165 .3383 .4871 .6631 .414 .736 I.I5 1.66 2.25 245 138 88.3 61.3 45 .1164 .2070 .3236 .4659 .6342 .396 .704 I.IO 1.58 2.16 268 151 96.5 67 49.3 ''H . .8661 1.096 2.94 3-73 34.5 27.3 .8284 1.048 2.82 3.56 37-7 30 1.353 1.637 1.949 4.60 5-57 6.63 22.5 18.3 153 1.294 1.566 1 1.864 4.40 5.32 6.34 24 20 16.8 t Oval-Flat Section. inch. inch. I x)^ .2510 .4463 .6974 .853 1.52 2.37 119 67 43 Table No. 93. — Weight of one Square Foot of Sheet Copper. To Wire-Gauge employed by Williams, Foster, & Co. Specific Weight taken as 1. 16 (Specific Weight of Wrought Iron = i). Thickness. Weight of! X Square i Foot Thickness. Weight of I Square Foot. Thickness. Weight of X Square Foot Wire- Gauge. No. Inch (approxi- mate). pounds. Wire- Gauge. No. Inch (approxi- mate). pounds. Wire- Gauge. No. Inch ^ (approxi- mate). pounds. I .306 14.0 II .123 565 21 .0338 1-55 2 .284 13.0 12 .109 5.00 22 .0295 1-35 3 .262 12.0 13 .0983 4.50 23 .0251 i-iS 4 .240 II. 14 .0882 4.00 24 .0218 I.OO 5 .222 10.15 15 .0764 3.50 25 .0194 .89 6 7 .203 .186 930 8.50 16 17 .0655 .0568 3.00 2.60 26 27 .0172 .0153 •79 .70 8 .168 7.70 18 .0491 2.25 28 .0135 .62 9 •153 7.00 19 .0437 2.00 29 .0122 •56 10 .138 6.30 20 .0382 1.75 30 .0110 •50 i WEIGHT OF METALS, Table No. 94. — Weight of Copper Pipes and Cylinders, BY Internal Diameter. Length, i Foot. Thickness by HoIuapHera Wite-Gauge (Table No. 13). Spedfic Weight-i.l6 (Specific Wdght of Wrought-Iron-i). Di^H. Ito. Ibt ibi. Ila. Ibi. Ita. lbs. Lbs. lbs. ibt. H 3- "4 2.84 2-33 1.92 1.53 I.4I 1.21 1.05 ■934 f?i % 3-84 3-49 2.91 2.44 1,99 1.84 1.60 1.41 1.27 H 4-54 4.13 3-49 2-95 2-45 2.27 1.77 1.60 1-43 'A 5-^3 4.78 4.06 3-47 2.91 2.71 2-39 2.13 •93 1-73 H 5-93 5.42 6.07 4.64 3-99 3-37 3-83 3- '4 2.7E It 2.26 2.04 6.63 5-22 4.50 3-57 3. "7 2.60 2.3s 7.32 6.71 5-79 S.02 4-29 4.00 3.57 3.22 2.93 2.66 ' 7.36 6.37 5-53 4.74 4.43 3-96 3-57 3-26 2.97 iH 8.71 8.00 6.95 6.05 S.20 4. 86 435 394 3.60 3.28 "X 9.4D 8,6s 7.52 6.57 ii\ S-29 4-75 4-30 m ]:g 'H 9.30 8.10 7.08 S-72 5-14 4.66 t'A lo's 9-94 8.68 7,60 6.57 6,16 5-53 S-02 4.60 4.20 'H ii.S 10.6 9.26 8.12 7.02 6-59 5-93 5-39 4-93 til iH 9.83 8.63 7.48 7.02 6.32 5-75 S-27 m ia!s 11.9 10,4 9-15 7.93 ;:si 6.71 6.11 5.60 S.I2 '3.5 12.S 9.66 8-39 7.11 6.47 5-93 5-43 2H 14.2 i3.a 11.6 10.2 8.84 8.31 6.83 6.27 5-74 i% 14.9 13-8 10.7 9-3° 8.74 7- "9 6.60 6.05 2H '5-6 14-5 12,7 9-75 9- '7 7-56 6.94 6.36 'H 16.3 IS- 1 13.3 niy 9.60 7.92 7.27 6.67 'H 17.0 158 •3-9 122 10.7 10.0 9.07 8. 38 7.60 ',% ^H 17.7 16.4 •4-5 ii^S 10.5 9-47 8.64 7.94 aji .§.4 17. 1 iS-o 13-3 13-8 II-S7 10.9 9.86 9.00 8.27 7- 59 3 "9-1 17-7 15.5 "■3 10.3 9.36 8.61 7.90 3J4 iO.4 19.0 16.8 148 12.9 ii,2 U.I 10.1 9.17 8.52 3H 21.8 17-9 •5-9 51 131 11.8 10.8 9-94 9-13 3K 23-2 21.6 19. 1 16.9 m 12.6 ii-s ia.6 9-75 4 24.6 22.9 17.9 '5-7 •3-4 12.3 "-3 10.4 4X 15.9 24.2 21.4 19,0 16.6 :« 14-Z 13-0 12.0 110 A'A It', r, 22.5 '^5 '5-2 '3-7 12.7 11.6 ^M 23-7- 24-8 18.4 17.4 IS.8 14.4 13-3 5 30-1 28,0 22.1 19.3 18.Z 16.6 IS-I 14.0 12.8 %%, 32.8 29-3 26.0 23-1 10.2 19. 1 17-3 15-9 14.6 '3-5 s'A 30.6 27.1 24.1 18. t 16.6 «S-3 141 SU 34.2 31-9 28.3 25.2 2a!8 .8.9 ;i:5 16.0 14.7 6 356 33-2 29.5 26.2 23.0 '■•' 19-7 16.6 '5-3 S.04, 5-32 WEIGHT OF COPPER PIPES AND CYLINDERS. 263 Table No 94 {contintied). Length, i Foot. Thickness by Holtzapffers Wire- Gauge (Table No. 13). Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i). Thick- kiss. 8 9 10 II la 13 14 15 16 17 18 19 ao W.G. Inch. .165 .148 •134 .120 .109 .095 .083 .072 .065 .058 .049 .042 •035 "/64* 9/6</ lbs. 9/64^ H^ 7/64 3/3a/ lbs. 5/64^ lbs. 'A6/ Vi6^ 3/64/ 3/64^ ^M IST. DlAM. ,bches. Ihs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. .581 .491 .422 •357 .310 .254 .210 .173 .150 .129 .104 .086 .068 X .832 .716 .626 .540 .476 .398 .336 .282 .249 .217 .178 .149 .121 H 1.08 .941 .830 .722 .641 •553 .462 •391 .348 •305 .253 .213 !228 a 1-33 I.I7 1.03 .904 .807 .687 .588 .500 .447 .393 .327 •277 H 1.58 1.39 1.24 1.09 .972 .831 •Z'4 .610 .545 .481 .402 .341 .281 )i. 1-83 1.62 1.44 1.27 1. 14 .975 .840 .719 .644 .570 .476 .404 •334 }i 2.09 1.84 1.65 1.45 1.30 1. 12 .966 .828 •743 .658 •550 .468 •387 I 2.34 2.05 1.85 1.63 1.47 1.26 1.09 .938 .842 .746 .625 .532 .440 iH ^:i^ 2.27 2.05 1.82 1.63 1. 41 1.22 1.05 .940 .834 .699 .596 •493 iH 2.49 2.26 2.00 1.80 1-55 1.34 1. 16 1.04 .922 •774 .659 •547 '^, 309 2.72 2.46 2,18 1.97 1.70 1.47 1.27 1. 14 I.OI .848 .723 .600 'K 3.34 2.94 2.67 2,36 2.13 1.84 1.60 1.38 1.24 1. 10 .922 •787 .653 •K 359 3-17 2.87 2.55 2.30 1.99 1.72 1.48 1-34 1. 19 •997 .851 .706 % 3.&4 3-39 3-07 2.73 2.46 2.13 1.85 1.59 1.43 1.27 1.07 •9^5 •759 ^a 4.09 3.62 3.28 2.91 2.63 2.27 1-97 1.70 ^53 1.36 1.15 .978 .812 2 434 3.84 3.48 309 2.79 2.42 2.10 1. 81 1.63 1.45 1.22 1.04 .865 t% 459 407 3.69 327 2.96 2.56 2.23 1.92 H^ 1.54 1.29 i.ii .919 i'4 484 429 3.89 3-45 3- 12 2.71 ; ^•35 2.03 1.83 1.63 1.38 1.17 .972 % 5-09 4.52 4.09 3-64 329 2.85 2.48 2.14 1-93 H' 1-45 1-23 1.03 1.08 ^}i 5-34 4.74 4.30 3.82 3-45 3.00 2.60 2.25 2.03 1.80 1.53 1.30 »H l-s 497 4.50 4.00 3-62 3.14 2.73 2.36 2.13 1.89 1.60 1.36 1. 13 ^H 519 4.71 418 3-79 3.28 2.86 2.47 2.22 1.98 1.68 1-43 1. 18 m 6.09 5.42 4.91 4.37 3-95 3-43 2.98 2.58 2.32 2.07 H^ 1.49 1.24 3 6-34 5.66 5.11 4-55 4.12 3.57 3" 2.69 1 2.42 2.16 1.82 1-55 1.29 ^ 6.8s 6.11 5.52 4.91 4-45 3.86 3.36 2.91 2.62 2.33 1.96 1.68 1.40 3^ 7.8s 6.56 5-93 5.28 4.78 4.15 3.61 3.12 2.82 ^•5J 2. 1 1 1.81 1.51 3¥ 7.01 6.33 5.64 5.11 4.44 3.87 3.34 3.01 2.68 2.26 1.94 1.62 4 8-35 7.46 6.74 O.OI 5.44 4.73 4.12 3.56 3-21 2.86 2.41 2.06 173 4)( 8.85 7-9» 7.14 6.37 5-77 5.02 4.37 3.78 341 3-<H 2.56 2.19 1.84 *^/ 9-35 8.36 8.81 7-55 6.74 6.10 5-30 4.62 400 3.61 3.21 HI 2.32 1.94 aH 9.85 7.96 7.10 6.43 5-59 487 4.22 3.80 3-39 2.86 2.45 2.05 5 10.4 9.26 8.36 7.46 6.77 5.88 5.^3 4.44 4.00 356 301 2.57 2.16 5^ 10.9 9.71 8.77 7.83 7.10 6.17 5.38 4.66 420 3-74 3. IS 2.70 2.27 5^ 11.4 ia2 9.18 8.19 7.43 6.46 I'M 4.88 4.40 392 3.30 2.83 2.38 I^ 11.9 10.6 9.58 8.56 7.76 6.7s 5.09 4.59 4.09 3.45 2.96 2.48 6 •M II. I 9-99 8.92 8.09 7.04 6.14 5.31 4.79 4.27 3-6o 3-09 2.58 262 WEIGHT OF METALS. Table No. 94. — ^Weight of Copper Pipes and Cylinders, BY Internal Diameter. Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight = 1. 16 (Specific Weight of Wrought-Iron= i). Thick- KESS. W. G. Inch. Int. DlAM. inches. H H n 1% i}i iH 2 2)i 2H 2H 2>^ 2H 2H 2H 3 3X 3'A 3H 4 A'A aH 5 5^ SA sH 6 0000 •454 39/64 lbs. 3H 3-84 4-54 5-23 5-93 6.63 7.32 8.02 8.71 9.40 10. 1 10.8 11.5 12. 1 12.8 >3.S 14.2 14.9 15.6 16.3 17.0 17.7 18.4 19. 1 20.4 21.8 23.2 24.6 25.9 273 28.7 30.1 3'-S 32.8 34.2 35-6 000 00 .425 '7/64/ .380 lbs. lbs. 2.84 3-49 4.78 2.33 2.91 3-49 4.06 5.42 6.07 6.71 7.36 4.64 5.22 5.79 6.37 8.00 8.65 9.30 9.94 6.95 7-52 8.10 8.68 10.6 II. 2 11.9 12.S 9.26 9.83 10.4 II.O 13.8 H-5 15.1 1 1.6 12. 1 12.7 13.3 15.8 16.4 17.1 17.7 13.9 '4.5 15.0 156 19.0 20.3 21.6 22.9 16.8 17.9 19. 1 20.2 24.2 25.4 26.7 28.0 21.4 22.5 23.7- 24.8 29.3 30.6 31-9 33-2 26.0 27.1 28.3 29.5 .340 11/32 lbs. 1.92 2.44 2.95 3-47 3-99 4.50 5.02 5-53 6.05 6.57 7.08 7.60 8.12 8.63 9.15 9.66 10.2 10.7 II. 2 11.7 12.2 12.8 13.3 13-8 14.8 15.9 16.9 17.9 19.0 20.0 21.0 22.1 23.1 24.1 25.2 26.2 .300 '9/64/ lbs. 1.53 1.99 2.45 2.91 3.37 3.83 4.29 4.74 5.20 5.65 6. 1 1 6.57 7.02 7.48 7.93 8.39 8.84 930 9-75 10.2 10.7 II. I 11.57 12.0 12.9 139 14.8 15.7 16.6 18.4 19.3 20.2 21. 1 22.1 23.0 .284 9/32/ lbs. 1. 41 1.84 2.27 2.71 3.14 3-57 4.00 4-43 4.86 5.29 572 6.16 6.59 7.02 7-4S 7.8g 8.31 8.74 9.17 9.60 lO.O 10.5 10.9 "•3 12.2 131 '3-2 14.8 15.6 16.5 17.4 18.2 19. 1 20.0 20.8 21.7 .259 X/ lbs. 1. 21 1.60 2.00 2.39 2.78 317 3-57 396 4-35 4.75 5.14 5-53 5-93 6.32 6.71 7.II 7.50 7.89 8.29 8.68 9.07 9.47 9.86 10.3 1 1. 1 11.8 12.6 »3-4 14.2 15.0 15.8 16.6 17.3 18. 1 18.9 19.7 .238 '5/64/ lbs. 1.05 I.4I 1.77 2.13 2.50 2.86 3-22 3.57 3-94 4-30 4.66 5.02 5.39 5.75 6. 1 1 6.47 6.83 7.19 756 7.92 8.28 8.64 9.00 936 10. 1 10.8 II. 5 12.3 13.0 137 14,4 15. 1 15-9 16.6 'Z-3 18.0 .220 7/3"/ lbs. .934 1.27 1.60 1.93 2.26 2.60 2.93 3-26 3.60 3-93 4.26 4.60 4.93 5.27 5.60 5-93 6.27 6.60 6.94 7.27 7.60 7.94 8.27 8.61 9.27 9-94 10.6 "•3 12.0 12.7 13.3 14.0 14.6 15-3 16.0 16.6 .203 '3/64 lbs. .809 1. 12 L43 1.73 2.04 2.35 2.66 2.97 3.28 3.58 3.89 4.20 4-Si 4.82 5.12 5-43 5.74 6.05 6.36 6.67 6.97 7.28 7.59 7.90 8.52 9- 13 9.75 10.4 II.O II. 6 12.2 12.8 13.5 14. 1 14.7 15.3 .180 3/16 b. lbs. .667 .941 1. 21 1.49 1.76 2.03 2.31 2.58 2.85 313 340 3.68 3.95 4.22 4.50 4-77 5.04 5-32 5.86 6.14 6.41 6.68 6.95 7.50 8.04 8.59 9.»3 9.67 10.2 10.8 "3 11.9 12.4 12.9 13-5 WEIGHT OF COPPER PIPES AND CYLINDERS. 263 Table No 94 {continual). Length, i Foot. Thickness by Holtzapffers Wire- Gauge (Table No. 13). Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i). Thick- KBSS. W. G. • 8 9 10 IZ 12 13 14 15 .072 5/64^ lbs. z6 .065 Vi6y 17 z8 Z9 20 Inch. .165 lbs. .148 9/64/ lbs. .134 9/64^ .120 .109 7/64 .095 3/3>/ 1 lbs. .083 lbs. .058 .049 3/64/ .042 3/64^ .035 Int. DiAM. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. inches. i .581 .832 1.08 1.33 .491 .716 .941 I.17 .422 .626 .830 1.03 •357 .540 .722 .904 .310 .476 .641 .807 •254 .398 '.687 .210 .336 .462 .588 •173 .282 •391 .500 .150 .249 .348 .447 .129 .217 •305 •393 .104 .178 .253 .327 .086 .149 .213 .277 .068 .121 •'75 .228 i I 1.58 1-83 2.09 2.34 1-39 1.62 1.84 2.05 1.24 1.44 1.65 1.85 1.09 1.27 1.63 .972 1. 14 1.30 1.47 .831 .975 1. 12 1.26 .714 .840 .966 1.09 .610 .719 .828 .938 •545 .644 .743 .842 .481 .570 .658 .746 .402 .476 .550 .625 •341 .404 .468 .532 .281 .334 .387 .440 2.59 2.84 3-09 3-34 2.27 2.49 2.72 2.94 2.05 2.26 2.46 2.67 1.82 2.00 2.18 2.36 1.80 1.97 2.13 I.4I 1.55 1.70 1.84 1.22 1-34 1.47 1.60 1.05 1. 16 1.27 1.38 .940 1.04 1.14 1.24 •834 .922 I.OI 1. 10 .699 .774 .848 .922 .596 .659 .723 .787 •493 .547 .600 .653 2 4.09 4-34 317 3-39 362 3.84 2.87 3-07 328 3.48 2.55 2.73 2.91 3-09 2.30 2.46 2.63 2.79 1.99 2.13 2.27 2.42 1.72 1.85 1.97 2.10 1.48 1.59 1.70 I.81 1-34 1.43 1.53 1.63 1. 19 1.27 1.36 1.45 •997 1.07 i.iS 1.22 .851 .978 1.04 .706 .759 .812 .865 2H 4-59 4.84 5.09 5.34 4.07 4.29 4-74 3-69 3.89 4.09 4.30 3-27 3-45 3-64 3.82 2.96 3.12 329 3.45 2.56 2.71 2.85 3.00 2.23 2.35 2.48 2.60 1.92 2.03 2.14 2.25 '•Z3 1.83 1-93 2.03 1-54 1.63 1.29 1.38 1.45 1-53 I. II 1.17 123 1.30 .919 .972 1.03 1.08 3 6.09 6.34 4.97 5.19 5.42 5.66 4.50 4.71 4.91 5.II 4.00 4.18 4.37 4.55 362 3-79 3.95 4.12 3- '4 3.28 3-43 3-57 2.73 2.86 2.98 3." 2.36 2.47 2.58 2.69 1 2.13 2.22 2.32 2.42 2.07 2.16 1.60 1.68 1.82 1.36 1-43 1.49 1.55 1. 18 1.24 1.29 3U 3% 3H 4 6.85 7.85 8.35 6.1 1 6.56 7.01 7.46 5.52 5.93 6.33 6.74 4.91 5.28 5.64 O.OI 4.45 4.78 5.11 5.44 3.86 415 4-44 4.73 3.36 3-6i 3.87 4.12 2.91 312 3-34 3-56 2.62 2.82 3.01 3.21 2.33 2.51 2.68 2.86 1.96 2. 1 1 2.26 2.41 1.68 1. 81 1.94 2.06 1.40 1.51 1.62 1.73 4X 4K 4H 5 8.85 9-35 9.85 10.4 7.91 8.36 8.81 9.26 7.14 7.55 7.96 8.36 6.37 6.74 7.10 7.46 6.10 6.43 6.77 5.02 5-30 5-5§ 5.88 4.37 4.62 487 5.13 3.78 4.00 4.22 4.44 341 3.61 3.80 4.00 3.04 3.21 3.39 356 2.56 2.71 2.86 3.01 2.19 2.32 2.45 2.57 1.84 1.94 2.05 2.16 10.9 11.4 11.9 12.4 9.71 ia2 10.6 II. I 8.77 9.18 9.58 9.99 7.83 8.19 8.56 8.92 7.10 7.43 7.76 8.09 6.17 6.46 6.75 7.04 5.38 6.14 466 4.88 5.09 5-31 4.20 4.40 459 4.79 3^74 392 4.09 427 3.15 330 3.60 2.70 2.83 2.96 3-09 2.27 2.38 2.48 2.58 264 WEIGHT OF METALS. Table No. 94 {continued). Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight=i.i6 (Specific Weight of Wrought- Iron =1). Thick- ness. 0000 000 00 z 2 3 4 5 6 7 W. G. •454 »6/64 .425 .380 .340 .300 .284 .259 .238 .220 .203 .180 Inch. '7/64/ ^/ "/33 19/64/ 9/3^/ x/ '5/64/ Vs"/ ^3/64 3/16 b. Int. • DiAM. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. inches. eVz 38.4 35.8 31.8 28.3 21.3 19.5 18.0 14.6 7 41. 1 38.3 34.1 30.3 26.6 25.1 22.8 20.9 19.3 17.8 15-7 VA 43-9 40.9 36.4 32.4 28.4 22.4 20.6 16.8 8 46.6 43-5 38.7 34.5 303 28.6 26.0 23.8 22.0 20.2 17-9 9 52.1 48.7 43-3 38.6 33.9 32.0 29.1 26.7 24.6 22.7 20.1 10 57.7 53.8 479 ^•2 37.5 35-5 32.3 29.6 273 25.2 22.2 11 ^2-^ 59-0 52.5 46.8 41.2 38.9 35-4 32.5 30.0 27.7 24.4 12 68.7 64.2 57.2 51.0 44.8 42.4 38.6 35.4 32.7 30.1 26.6 13 74.2 69.3 61.8 55.1 48.5 45.8 41.7 38.3 35.3 32.6 28.8 14 79.7 74.5 66.4 59-2 52.1 49.3 44.9 41.2 38.0 35-1 31.0 15 85.2 P't 71.0 63.4 55.8 52.7 48.0 44.1 40.7 37.6 33-2 16 90.7 84.8 75.6 67.7 59^4 56.2 51.2 46.9 43-4 40.0 35-4 17 96^3 90.0 80.2 71.8 63.0 59.6 54.3 49.8 46.0 42.5 37.5 18 101.8 95- 1 84.9 76.0 66.7 63.1 57.4 52.7 48.7 45.0 39.7 19 107.3 100.3 89.5 80.1 70.3 66.5 60.6 5g.6 51.4 47.4 41.9 20 112.8 105.5 94.1 84.2 74.0 70.0 63.7 58.5 54.0 49-9 44.1 21 118.3 1 10. 7 98.7 88.3 77.6 73.4 66.9 61.4 56.7 52.4 46.3 22 123.8 115.8 103.3 92.5 81.3 76.9 70.0 64.3 59.4 54.9 48.5 23 129.3 120.9 107.9 96.6 84.9 80.3 73-2 67.2 62.1 57.3 50.7 24 I34^8 1 26. 1 112.6 100.6 88.6 83.8 76.3 70.1 64.7 59.8 52.9 26 146.0 136.4 121. 8 108.8 95.9 90.7 82.6 759 70.1 64.7 57.2 28 ^IZ-* 146.7 131-0 117.1 103. 1 97.6 89.0 81.7 Z5-4 69.7 61.6 30 168.4 157.1 140.2 125.4 1 10.4 104.5 95.3 87.5 80.8 74.6 66.0 32 179.6 167.4 149.5 133.6 117.7 111.4 101.6 93.3 86.2 79.6 70.4 34 190.7 ^IV 158.7 141.9 125.0 1 18.3 107.9 99.1 91.5 84.5 74-7 36 201.9 188.0 167.9 150. 1 132.3 125.2 114.2 104.9 96.9 89.5 79.1 WEIGHT OF COPPER PIPES AND CYLINDERS. 265 Table No. 94 {continued). Length, i Foot. Thickness by Holtzapffers Wire-Gauge (Table No. 13). Specific Weight = 1. 16 (Specific Weight of Wrought Iron= i). Thick- mess. 8 9 10 zz Z2 13 14 15 16 17 18 19 20 W. G. Inch. .165 .148 .134 .120 .109 .095 .083 .072 .065 .058 .049 .042 •035 "/fi*^ 9/64/ 9/64^ 7/64 3/3>y sM 5/64^ V.6/ V16* 3/64/ 3/64* V3«/ Int. DiAM. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. inches. e>% 13.4 12.0 10.8 9.65 8.75 7.61 6.64 5.75 5- '2 462 3-90 3-34 2.80 7 14.4 12.9 II. 6 10.4 9.42 8.19 7.14 6.19 5.58 497 420 3.60 301 2^ 15.4 13.8 12.47 II. I 10. 1 8.77 7.65 6.63 5.98 5.33 449 3.85 3-23 8 16.4 147 13-2 II. 8 10.74 9.34 8.15 7.06 6.37 5.68 479 4.10 3.43 9 18.4 'f-5 14.9 ^3-3 12. 1 10.5 9.16 7.94 7.16 6.38 5.39 4.61 3.86 10 20.4 18.2 16.5 148 13.4 II. 7 10.2 8.81 7.95 7.08 5.98 5.12 4.28 II 22.4 20.0 18. 1 16.2 147 12.8 II. 2 9.69 8.74 7-79 6.58 5.63 470 12 24.4 21.8 19.8 17.7 16.0 140 12.2 10.6 9.53 8.49 7.18 6.14 5.13 13 26.4 23.6 21.4 19. 1 'Z-4 15.1 132 11.4 10.3 9.20 7.77 6.65 5.55 14 28.4 1 25.4 23.0 20.6 18.7 16.3 14.2 12.3 II. I 9.90 In 7.16 5.98 15 30.4 27.2 24.6 22.1 20.0 17.4 15.2 13-2 II.9 10.6 8.96 7.67 6.40 16 32.4 29.0 26.3 23.5 21.3 18.6 16.2 141 12.7 "3 9.56 8..18 6.82 17 34-4 30.8 27.9 25.0 22.7 19.7 17.2 149 13.5 12. 1 10.2 8.69 7.27 18 36.4 32.6 29.5 26.4 24.0 20.9 1 18.2 1' 15.8 14.3 12.7 10.7 9.20 7.69 19 384 34.4 31.2 27.9 25.3 22.0 19.2 16.7 I5.I 13-4 II-3 9.71 8.12 20 40.4 36.2 32.8 29-3 26.6 23.2 20.2 17.6 159 14. 1 11.9 10.2 8.54 21 42.4 38.0 34.4 30.8 27.9 24.3 21.3 18.4 16.6 14.8 12.S 10.7 8.96 22 4*.4 39.8 36.0 32.3 29-3 25.5 22.3 19.3 17.4 15-5 131 II. 2 9.39 23 46.4 41.6 37.7 33.7 30.6 26.7 233 20.2 18.2 16.2 137 11.8 9.81 24 48.5 43-4 39.3 35.2 31-9 27.8 24.3 21. 1 19.0 16.9 143 12.3 10.2 26 52.5 47.0 42.6 38.1 34.6 30.1 26.3 22.8 20.6 18.4 15.5 133 II. I 28 56.5 50.6 45.8 41.0 37.2 32.4 28.3 24.6 22.2 19.8 16.7 143 11.9 30 60.5 54.2 49.1 43.9 39.9 347 30.3 26.3 237 21.2 »7.9 15.3 12.8 32 ^^ 57.8 52.3 46.8 42.5 37.0 32.3 28.1 25.3 22.6 19. 1 16.3 13.6 34 68.5 61.4 55.6 49.8 ^5-1 39.4 344 29.8 26.9 240 20.3 17.4 145 36 1 1 1 72.5 65.0 58.8 52.7 47.8 41.7 36.4 31.6 28.5 25.4 21.5 18.4 15.3 266 WEIGHT OF METALS. Table No. 95. — ^Weight of Brass Tubes, BY External Diameter. Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight=i.ii (Specific Weight of Wrought Iron=i). Thick- NBSS. W. G. 15 z6 17 18 19 20 ax aa 23 24 25 vInch. .072 5/64^ .065 .058 '/16 b. .049 3/64/ .042 3/64 b. .035 V32/ .032 ' V32 .028 1/32 b. .025 1.6/64 .022 1.4/64 .020 1.3/64 DiAM. inches. lbs. lbs. lb. lb. lb. lb. lb. lb. lb. lb. lb. 3/16 s/16 .201 .187 .087 .130 .172 .079 .115 .150 .072 .102 .132 .037 .062 .088 ."3 .o3| !o8i .104 .031 .052 .072 .092 .029 .047 .065 .083 .026 .042 .058 .074 .024 •039 .053 .068 7/16 9/16 .306 .358 .411 .234 .281 .329 •376 .214 .256 .298 .340 .186 .221 .257 .293 .163 •»93 .224 .254 .138 .164 .189 •215 .128 .151 .174 .197 ."3 •133 •154 .174 .102 .120 .138 .156 .090 .106 .122 .138 .082 .097 .III .126 ^3/16 •463 .515 .567 .620 .423 .470 .517 .564 .382 •424 .467 .509 .328 .364 •399 .435 .285 .346 .376 .240 .265 .291 .316 .221 •244 .267 .290 .194 .215 •235 •255 .174 .192 .211 .229 .154 .170 .186 .202 .141 .155 .170 .184 »5/i6 I .672 .724 .'S8i .611 .658 .706 .801 •S5I .593 .635 .719 .471 .506 .542 .613 .407 .437 .468 .529 .342 •367 •393 .443 .314 •337 .360 .407 .276 .296 .316 •357 .247 .265 . .283 .320 .218 .234 .250 .282 .199 .213 .228 •257 .986 1.09 1.20 .896 .991 1.09 .972 .684 .827 .590 .651 .712 .494 .545 .596 .453 .546 .398 .439 .479 .356 .392 •429 •314 .346 .378 .286 •315 .344 W. G. 9 ID XX 12 13 14 15 16 17 18 19 Inch. .148 9/64/ lbs. .134 9/64^. .120 yib. .109 7/64 .095 3/3a/ .083 s/64/ .072 5/64 b. .065 .058 x/x6 b. .049 3/64/ .042 3/64*. DiAM. inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 1.90 2.II 2.33 2.54 1.74 1.93 2.13 2.32 '•5f 1.76 1.94 2.12 1.45 1.60 1.76 1.92 1.28 1. 41 1.55 1.69 113 1.25 1-37 1.49 .986 .991 1.20 1.30 .896 .991 1.09 1. 18 .804 .888 •972 1.06 .684 •755 .827 .898 .590 .651 .712 .773 2 2>^ a. 76 2.97 319 3-40 2.52 2.71 2.91 3.10 2.30 2.47 2.65 2.83 2.08 2.24 2.39 2.55 1.83 1.97 2.10 2.24 1. 61 1.85 1.97 1.40 1. 61 1.72 1.28 ^•37 1.56 1.14 1.23 1-31 139 .969 1.04 i.ii 1.18 .834 •895 .956 1.02 2X 2^ 2^ 3.62 3.83 4.04 3.30 3.49 3.69 3.01 319 3.37 2.71 2.86 3-02 2.38 2.52 2.66 2.09 2.21 2.33 1.82 1.93 2.03 1.66 1.85 1.48 1.56 1.65 1.25 133 1.40 1.08 1. 14 1.20 WEIGHT OF BRASS TUBES. 267 Table No. 95 (continued). Length, i Foot. Thickness by Holtzapffel's Wire-Gauge (Table No. 13). Specific Weight = I. H (Specific Weight of Wrought Iron=i). Thick- NBSS. W. G. 3 4 5 6 7 8 9 ID IX Z2 13 .259 .238 .220 .203 .180 .165 .148 .134 .120 .109 •095 Inch. x/ '5/64/ 7/3>/ '3/64 3/16 ^. "/64 *• 9/64/ 9/64^. >^^. 7/64 3/32 DiAM. inches. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 2 5.24 4.87 ^55 4.24 3.80 3.52 3.19 2.91 2.65 2.39 2.10 2yi 5.62 5-22 4.87 4-54 4.07 3-76 3.40 3-«o 2.83 2.55 2.24 2H 5-99 5-57 519 4.83 4-33 4.00 3.62 3-30 3.01 2.71 2.38 2H 6.37 591 551 5.13 459 4.24 3.83 3-49 3- 1-9 2.86 2.52 ^H 6.75 6.26 5-83 5.42 4.85 4.48 4.04 369 3-37 3- 02 2.66- ^^ 7.12 6.60 6.14 5.72 5-"> 4.72 4.26 3.88 3-55 3.18 2.79 2^ 7.50 6.95 6.47 6.01 5.38 4.96 4.47 4.07 3-73 3-34 2.93 ^H 7.88 730 6.79 6.31 5.64 5.20 4.69 4.27 391 3.50 307 3 8.25 7.64 7.II 6.60 590 S-44 4.90 4.46 4.09 3.66 3.21 3V 9.01 8.33 7.75 7.19 6.43 5.92 5.49 4.85 4.43 3.98 3.48 3>^ 9.76 9.02 8.39 7.78 6.95 6.40 6.07 5-24 4.78 4-30 3.76 3H 10.5 9.72 9.03 8.37 7-47 6.88 6.65 '5-63 5.12 4.61 4.04 4 "•3 10.4 9.67 8.96 8.00 7.36 7.24 6.02 5.46 4.93 4.31 4^ 12.0 ii.i 10.3 955 8.52 7.83 7.82 6.41 5.80 5.25 4.59 ^H 12.8 11.8 10.9 10. 1 9.04 8.31 8.41 6.80 6.15 5§S 4.87 aH 13.5 12.5 II.6 10.7 9.56 8.79 8.99 7.19 H^ 5.88 514 5 14.3 13-2 12.2 "3 10. 1 9.27 9-57 7.58 6.83 6.20 5.42 5^ ^^•?. X3-9 12.9 11.9 10.6 9.75 10.2 7.97 7.17 6.51 5.69 ^'4 15.8 14.6 13.5 12.5 II.I 10.2 10.7 8.36 7.52 6.83 5-97 sH 16.5 15-3 14. 1 131 II. 7 10.7 "3 8.75 7.86 7.1S 6.25 6 17.3 159 14.8 13.7 12.2 II. 2 11.9 9.14 8.20 7.46 6.52 268 WEIGHT OF METALS. Table No. 96. — Weight of One Square Foot of Sheet Brass. Thickness by Holteapffeirs Wire-Gauge (Table No. 13). Weight of Weight of Weight of Thickness. I Square Thickness. X Square Thickness. T Square Foot. Foot. Foot. No.W.G. inch. pounds. N0.W.G. inch. pounds. N0.W.G. ■inch. pounds. 3 •259 10.9 II .120 5-05 19 .042 1.77 4 .238 lO.O 12 .109 4.59 20 •035 1.47 5 .220 9.26 13 •095 4.00 21 .032 1-35 6 .203 8.55 14 .083 3-49 22 .028 i.i8 7 .180 7.58 15 .072 3.03 23 .025 1-05 8 .165 6.95 16 .065 2.74 24 .022 .926 9 .148 6.23 17 .058 2.44 25 .020 .842 10 .134 SM 18 .049 2.06 Table 97. — Size and Weight of Tin Plates. Mark. Size of Sheets. Number of Sheets in a Box. Weight per Sox. IC IX IXX IXXX IXXXX inches, inches. 14 X 10 » >l if >» it If sheets. 225 >» 99 pounds. 112 140 161 182 203 SDC SDX SDXX SDXXX S D XXXX 15 X II >f 9f » if » » 200 168 189 210 231 252 DC DX DXX DXXX DXXXX 17 X 12}^ 100 98 126 147 168 169 WEIGHT OF TIN AND LEAD PIPES. 269 Table No. 98. — ^Weight of Tin Pipes, As manufactured. I FOOT IN LENGTH. Diameter Externally. inches. 1/ n H 1 1 ■i 2 Thickness. 3/32" inch. lbs. .148 .384 .620 .856 1.095 1.328 1.564 1.802 ^ inch. lbs. .472 .787 T.IO3 I.417 1-732 2.047 2.362 Diameter Externally. inches. 2l' 2^ 3 Thicknbss. }i inch. lbs. 5-04 5-67 6.30 6.93 7.56 8.19 Table No. 99.— Weight of Lead Pipes. As manufactured. ! Bore. Length. Weight and Thickness of Pipe. Calcu- Calcu- Calcu- Calcu- Weight. lated Thick- ness. 1 Weight lated Thick- ness. Weight. lated Thick- ness. Weight lated Thick- ness. inches. feet lbs. inch. lbs. inch. lbs. bch. lbs. inch. >^ IS 14 .097 16 .112 18 .124 22 .146 ^ n 17 .101 21 .121 30 .140 Va w 24 .112 28 .147 32 .181 36 •215 I » 36 .136 1 42 .156. 56 .200 64 .225 ^Va 12 36 .139 42 .160 48 .180 52 •193 ^% » 48 .156 56 .179 72 .224 84 .257 ^H w 72 .199 84 .228 96 .256 2 » 72 .178 84 1 .204 96 .231 112 .266 2^ 10 i 84 .200 96 .227 112 .261 3 » 112 .218 140 •275 3J^ » 130 .225 160 •273 4 » 170 .257 220 •327 4j^ 19 170 .232 220 •295 4H, 4'A V.6ii ich thick. W< sight pel 9J r lineal f bot .... 22.04. lbs. 99 .... 23.25 ff 4H 5 99 99 .... 2*1. ( 56 „ 99 -TV jy 270 WEIGHT OF METALS. Table No. loo. — Dimensions and Weight of Sheet Zinc. ( Vielie-Montagne.) SiZBS OP Shebts. Weight No. Thickn«»« 9.0 X. 50 metres; 3.0 X. 65 metres; 9. ox. 80 metres; per area, x square metre. area, 1.3 sq., metres. 6. 56 X 2. X3 feet; area. area, 1.6 sq. metres. 6 '56X9. 62 feet; area. square toot. 6. 56 X X. 64 feet ; area, X0.76 square feet. 13.99 square feet. X7. 29 square feet. No. inxn. inch. kUs. lbs. kUs. lbs. kils. lbs. lbs. 9 .41 .0161 2.90 6.39 3.70 8.16 4.6 10.14 .589 lO •51 .0201 3-45 7.61 4.45 9.81 5.5 12.12 .704 II .60 .0236 4.05 8.93 5.30 11.68 6.5 14.33 .832 12 .69 .0272 4.65 10.25 6.10 13.45 7.5 16.53 .960 13 .78 .0307 5-30 11.68 6.90 15.21 8-5 18.74 1.088 14 .87 •0343 5.95 13-12 7.70 ^5*§^ 9-5 20.94 I.216 ^5 .96 .0378 6.55 14.44 8.55 18.85 10.5 23.15 1.344 i6 1. 10 .0433 7.50 16.53 9-75 21.50 12.0 26.46 1.536 17 1.23 .0485 8.45 18.63 10.95 24.14 13.5 29.97 1.740 i8 1.36 .0536 9-35 20.61 12.20 26.90 15.0 33.07 1.920 19 1.48 .0583 10.30 22.71 13.40 29.54 16.5 36.38 2.II2 20 1.66 .0654 11.25 24.80 14.60 32.19 18.0 39.68 2.304 21 1.85 .0729 12.50 27.56 16.25 35-82 20.0 44.09 2.560 22 2.02 .0795 13.75 30.31 17.90 39.46 22.0 48.50 2.816 23 2.19 .0862 15.00 33-07 19.50 42.99 24.0 52.91 3.073 24 2.37 •0933 16.25 35.82 21.10 46.52 26.0 57.32 3.329 ^5 2.52 .0992 17.50 38.58 22.75 50.15 28.0 61.73 3.585 26 2-66 .1047 18.80 41.44 24.40 53.79 31.0 68.34 3.969 Table No. loo {continued). Special Sizes for Sheathing Ships. Sizes op Sheets. Weight No. Thickness. X.1SX.35 metres; 1.30 X. 40 metres; per area, .402 sq. metre. ] 3.77 X X. 15 feet; area, 4-33 sq- feet. area, .520 sq. metre. 4. 26X1. 31 feet; area, 5.60 sq. feet. square foot. No. mils. inch. i kUs. lbs. kils. lbs. lbs. 15 .96 .0378 2.65 5.84 3.40 W 1.344 16 1. 10 .0433 3.00 6.61 3.90 8.60 1.536 17 1.23 .0485 3.40 7.50 4.40 9.70 1.740 18 1.36 ■0536 3.75 8.27 4.90 10.80 1.920 19 1.48 .0583 , 4.15 9.15 5.35 11.79 2.II2 20 1.66 .0654 ; ^55 10.03 5.85 12.90 i 2.304 1 Note. — A deviation of 25 dekagrammes, or about half-a-pound, more or less, from the proper weight of each number of sheet, is allowed. Nos. I to 9 are employed for perforated articles, as sieves, and for articles de Paris, Nos. 10 to 12 are used in the manufacture of lamps, lanterns, and tin-ware generally, and for stamped ornaments. The last numbers are used for lining reservoirs, and for baths and pumps. FUNDAMENTAL MECHANICAL PRINCIPLES. FORCES IN EQUILIBRIUM. Solid Bodies. Paralldogram of Forces. — ^When a body remains at rest whilst being acted on by two or more forces, it is said to be in a state of equilibrium, and so also are the forces. Thus, if the forces p/, q ^, r r, Fig. 86, acting on the body pqr^ keep it at rest, they are in equilibrium, and any two of them balance the third. The lines of force, if produced, meet at one point o within the body, and if a parallel- r (^ram be constructed having two adjacent sides proportional to and parallel to two of the forces respec- Fig. 86.— EquiUbrium of Forces. tively, to represent them in magni- tude and direction, the diagonal of the parallelogram will represent the third force in magnitude and direction. Let the lines o p, OQ, Fig. 87, represent the forces p/, q^ in magni- tude and direction, and com- :p plete the parallelogram • by drawing the parallels p r, q r, Bf_„ and draw o r. Then o r re- presents in magnitude and direction the resultant of the ^ two forces; and RO taken in Fig. 87.— Parallelogram of Forces. the opposite direction repre- sents the third force Rr, Fig. 86. If it be applied in this direction to the point o, as indicated by a dot line o r', it would balance the other two. This construction is called the paralldogram of forces y and is applicable to any three forces in equilibrium. Three forces in equilibrium may also be represented by a triangle, or half a parallelogram. For example, the triangle o p r represents by its three sides the forces o P, o q, o r, the side p r being substituted for o q. Three forces in equilibrium must be in the same plane. When the directions of three forces holding a body at rest, and also the magnitude of one of them, are known, the magnitudes of the other two can be determined by constructing a parallelogram in the manner above exem- plified, and measuring the lengths of the 5ide^ apd the diagonal. 2/2 FUNDAMENTAL MECHANICAL PRINCIPLES. Polygon of Forces, — Equilibrium may subsist between more than three forces, which need not necessarily be in the same plane; and they can, like those already illustrated, be developed in direction and magnitude by diagram. Thus, let the point o, Fig. 88, representing a solid body, be kept at rest by a number of forces, op, o q, o r, o s, o t. Find the equivalent diagonal o/ for the first two forces ; then construct the parallelogram and diagonal o r for the resultant of op and the third force or; and again the parallelogram and diagonal o s for the resultant of o r and the fourth force o s. The last resultant o^ represents in one the four distributed forces op, oq, or, os, and it balances the fifth force o t equal and opposite to it. A5 o j and o t are in the same straight line, their resultant is of course nothing. The several forces thus dealt with may be combined into a polygon of forces. Draw o p. Fig. 89, parallel and equal to o p, Fig. 88, p q parallel and equal to o Q, Q r parallel to o r, r s parallel to os; then, finally, so, completing the polygon, will be parallel and equal to ot. Fig. 88, the last of the series. Professor Mosely illustrates the polygon of forces by the united action of a number of bell-ringers, pulling by a number of ropes joined to a single rope. The polygon constructed as in Fig. 90, shows successively by corresponding letters, the individual contributions of the bell-ringers, combined into one vertical force. Again, equilibrium may be estab- lished between a number of forces Fig. 88.— Equilibrium of more than Three Forces. acting in the same plane, applied to different points in a body, or system of bodies. For example, let the forces p o, q o, r o, s o, and t o, be applied to several points, o, o, o, o, o, on a flat board ABC, Fig. 91, by means of cords and weights; it will settle into a position of equilibrium, when the opposing forces arrive at a balance between themselves. An axis or pivot may be established at any point, m, on the surface of the board, without disturbing the equilib- rium, and it may be viewed as a centre of motion round which the forces tend to turn the board, some in one direction, the others the opposite way, balancing each other. The effect of each force to turn the body about the centre is represented by the product of its magnitude by the leverage^ or perpendicular distance of its direction from the centre; draw these perpendiculars, and multiply each force by its perpendicular or leverage, then the resulting products will be divisible into two sets, tending to turn the board in opposite directions. The sum of the first set of products is equal to the sum of the second set, as is proved by the fact of equilibrium. Moments of Forces, — ^The product of a force by the perpendicular dis- tance of its direction from any given point, is called the momait of the Fig. 89. — Polygon of Forces. FORCES IN EQUILIBRIUM. 273 force about that point; and the equilibrium above discussed, in connection with Fig, 91, is the result of the equality of moments. The law of the equality of moments may be thus set forth; — If any number of forces acting anywhere in the same plane, on the same body or connected system of bodies, be in equilibrium, then the sum of the forces tending' to turn the system in one direction about any point in that plane, is equal to the sum of the mo- ments of those forces tending to turn the system in the other direction. Such balanced forces may be transferred to a single point, and placed about it, as in l~ig. 88, parallel to their directions as they stand; and they will continue in equilibrium, holding the point at rest. A polygon of the iotctspgrsl within Fig. 90, may be constructed similarly to Fig. 89. Though the principle of the polygon of forces be sufficient to test the equilibrium of a system of forces acting at one point, yet the principle of the equality of moments, Fig. 90— Beu-nngen, Polygon of Forcei. in addition, is necessary to establish the equilibrium of a system applied to different points. The two principles conjointly are necessary, and they are sufficient, as conditions of equilibrium. The Catenary. — When a chain, or a rope, or a flexible series of rods, is suspended by its extremi- ties, supporting weights distributed along its length, in a state of rest, ii forms a polygon of forces in equilibrium, as in Fig. 92. If all the forces except those which o act on the extremities of the chain, be combined into a resultant, then the two extreme sides being produced, will meet the direction of the resultant at one point Thus, in the polygon, Fig. 92, loaded with weights, w,w, &C., the verrical resultant Fig. 91.— Equjiiry of Moments. of these weights w' W, passing through their common centre of gravity, intersects at w' the two extreme sections p a, p' b, when produced downwards. Similarly, in the catenary, Fig. 93, which is the curve assumed by a rope or other flexible medium uniformly loaded and suspended by the two 274 FUNDAMENTAL MECHANICAL PRINCIPLES. extremities, if tangents be drawn to the extremities a, b, of the curve, meeting at w*, they represent the directions of the forces sustaining the curve at Fig. 9a.— The Catenary. Fig. 93. — ^The Catenary. those points, and they intersect at the same point w*, the vertical line G w^ passing through tlie centre of gravity of the curve. Let the weight of the •^ flf T Fig. 94.— Centrifugal Forces in Equilibrium. Fig. 95.—- Parallel Forces in Equilibrium. curve be represented by g w^, and complete the parallelogram m n, then w* M and w*' n represent in force and direction the tension at the points B and A. Centrifugal Forces in Equilibrium. — ^\Vhen a cylindrical vessel is exposed to a uniform internal pressure, as the pressure of steam within a boiler, for example, the pressure is balanced by the resistance of cohesion of the material of the boiler. Let a b c D, Fig. 94, be the section of a cylindrical boiler, the radial pressure of the steam may be represented by the arrows, which are equal and opposite in direction. The tension on the metal in resisting the internal pressure at any particular section b i>, is equal to the sum of the pressures resolved into the direction at right angles to b d, or parallel to ac, according to the triangles, or half-parallelograms of force attached to each oblique arrow. The total vertical pressure thus obtained by the resolution of forces is equal to the total vertical pressure which FORCES IN e;quilibrium. 27s would be exerted on the sectional line B d if it be supposed to be a rigid diaphragm across the boiler, which is easily calculated. If the radial pressure be, for example, 100 lbs. on each square inch of surface, then the total pressure, or the tension on the two sides at b and d, would be 100 X BD on each inch of length of the two sides; that is to say, if the diameter b d be equal to 60 inches, the tension on the two sides would be 60 X 100=6000 pounds for each inch of length. A similar argument applies to the tension on the rim of a revolving fly- wheel. Parallel Farces. — Systems of parallel forces are particular cases of the forgoing. — Let a, b, c, d, e, f, Fig. 95, be a system of parallel forces in equili- brium; and MN a line perpendicular to them in the same plane, and cut by them at the points a^byCyd^e,/, They may act at any points in their lines of direction without disturbing the equilibrium, and they may be sup- posed to be applied at those points in the line m n. Then, the sum of the moments of the three forces a, b, c, acting in one direction, with respect to any point m as a centre, is equal to the sum of the moments of the forces d, e, f, opposed to them. Further, the sum of the simple forces A, B, c, irrespective of their moments, is equal to that of the forces d, e, f, so that the fact of their being in equilibrium resolves itself into a case of action and reaction, for the two equivalent forces representing the two opposing sums, act in the same straight line in opposite directions. When three parallel forces balance each other, acting on a straight line, two of them must be opposed to the third; and the third must act between the other two, being equal and opposite to their resultant. Let a, b, c, Fig. 96, be three such forces applied to the line e g f, at the points E, G, f respectively; then, with respect to the point G, the moment of the force b is nothing, because it passes through that point and has no leverage on it There remain the moments of the extreme forces, a and c, which are equal to each other, that Is to say axeg = cxfg. Fig. 96.— Three Parallel Forces in Equilibriuia. From this it follows, by proportion, that A : c : : FG : eg, and that the extreme forces are to each other inversely as their distances from the middle force. In general, of three parallel forces acting in equilibrium on an inflexible line, the first in order is to the third as the distance of the third from the second, is to that of the first from the second. The sum of the first and third is equal to the second; and when the distances or leverages are equal, the first and third forces are equal to each other. If the position of the line e f be inclined to the direction of the three forces, and changed to e' f'. Fig. 96, the forces A, b, c, continue in equilibrium; 2'j(> FUNDAMENTAL MECHANICAL PRINCIPLES. Fig. 97. — Parallelopiped of Forces. for the perpendicular lines g e and g f continue, as before, to be the lever- ages of the extreme forces a and c, on the central point g. When three forces not in the same plane act on one point, there cannot be equilibrium between thenL Two of these may be reduced to their resultant by parallelogram, and this resultant reduced with the third force to a final resultant For example, let the lines op, OQ, OR, Fig. 97, represent in magnitude and direction three forces not in one plane acting on the point o. By parallel- ogram, o s is the resultant of the two forces o p, o Q, and o t is the final resultant of o s and the third force o R. That is to say, o t is the resultant of the three given forces. If parallelograms be formed from each two of the three forces, they form, when duplicated, a parallelopiped of forces, of which the diagonal is found by the final resultant o t, and the principle of the parallelopiped of forces may be thus defined: — If three forces be represented in magnitude and direction by three adjacent edges of a parallelopiped, their resultant is represented in magnitude and direction by the adjacent diagonal of the solid. There must be at least four forces to produce equilibrium about a point, when the forces are not in the same plane. The triangle ost. Fig. 97, comprises in its three sides the resultant of the first and second forces, the third force, and the resultant of the three. If the first resultant o s be replaced by the two lines o Q and Q s, which represent the first and second forces, they form the four-sided figure o Q s t, the polygon of the four equilibrating forces. A greater number of forces than four acting on a point may be reduced in like manner. Fluid Bodies. The characteristic property of fluids is the capability of transmitting the pressure which is exerted upon a part of the surface of the fluid, in all directions, and of the same intensity: — the same pressure per square inch or per square foot. The pressure of water in a vessel, caused by its own gravity, increases in proportion to the depth below the surface; and the pressure on a horizontal surface, say, the bottom, is equivalent to the weight of the superincumbent column of water, and the intensity of the pressure is independent of the form of the vessel. The same rule applies when the pressure is from below upwards. The same rule also applies when the surface is either vertical or inclined, and the mean height of the superincumbent column of water is measured by the depth of the centre of gravity of the given surface below the siurface of the water. The water in open tubes communicating with each other, when in a state of equilibrium, stands at the same level in the tubes, whatever may be the relative diameters of the tubes. MOTION. — GRAVITY. 277 The height of the superincumbent column of water is called the head of water. The buoyancy^ or the upward force with which water presses a body- immersed in it, from below upwards, is equal to the weight of water dis- placed by the body, or of a quantity of water equal in volume to the sub- merged body, or submerged portion of a body. The resultant of the upward pressure passes through the centre of gravity of the water displaced; and also, when the floating body is at rest, through the centre of gravity of th« body. This resultant line is called the axis of floatation, and the horizontal section of the body at the surface of the water is \ht plane of floatation. Bodies float either in an upright position or in an oblique position. A body floats with stability , when it strives to maintain the position of equili- brium, and when it can only be moved out of this position by force, and will return to it when the force is withdrawn. The metacentre is the point at which the axis of floatation intersects the axis of a symmetrical body, as a ship, when inclined. If the metacentre lies above the centre of gravity of the ship, the ship floats with stability; if below, the ship is unstable; if the centres coincide, which they must do in a cylinder or a sphere, for example, the body floats indifferently in any position. For the weight, volume, and pressure of water and air, see Water and Air as standards of measure^ page 124. MOTION. The motion of a body is uniform, when the body passes through equal spaces in equal intervals of time. Velocity is the measure of motion, and is expressed by the number of feet or oflier unit of length moved through per second or other unit of time. Motion is accelerated, when the body moves through continually increased spaces in equal intervals of time, like a railway train starting from a station. Motion is retarded, when the body moves through continually decreased spaces in equal intervals of time, like a railway train arriving at a station. The acceleration and retardation are uniform, when the spaces moved through increase or decrease by equal successive amounts, like a body falling by the action of gravity, or, on the contrary, projected upwards in opposition to gravity. GRAVITY. When bodies fall freely near the surface of the earth, the motion, as already said, is uniformly accelerated; equal additions of velocity being made to the motion of the body in equal intervals of time. During the ist second of time, the body, starting from a state of rest, falls through 16.095 ^^^t, or, say 16.1 feet; during the 2d second, it falls through three times 16.1 feet; during the 3d second, it falls through five times 16.1 feet, and so oa The body having, in the ist second, fallen through 16. 1 feet, from a state of rest, with a motion uniformly accelerated, it would move through 32.2 feet in the next second, with the velocity thus acquired, without any additional stimulus from gravity; that is to say, the velocity acquired at the end of the ist second is 32.2 feet per second. During the 2d second, it in fact acquires an additional velocity of 32.2 feet per second, making up, at the end of this second, a final velocity of 64.4 2/8 FUNDAMENTAL MECHANICAL PRINCIPLES. feet per second. In like manner the body acquires an additional velocity of 32.2 feet per second during the 3d second, making a final velocity of three times 32.2 feet, or 96.6 feet per second. And so on. Each of these additional velocities is acquired in falling through 16.1 feet additional to the space fallen through in virtue of the velocity acquired at the beginning of each second. The relations of height fallen^ velocity acquired^ and time of fally are simply exhibited in the following manner : — During the successive seconds the heights fallen through are consecutiv|?ly as follow: — time, I, I, I, I second, height of fall, 16. i, 16. i x 3, 16. i x 5, 16.1 x 7 feet And reckoning the totals from the commencement of the fali, total times, i, 2, 3, 4 seconds, total height of fall^ 16.1, 16.1 x 4, 16. i x 9, 16. i x 16 feet. or t6.i, 16. 1 X 2^, 16. 1 X 3^, 16. i x 4^ feet. or 16.1, 64.4, i44'9> 257.6 feet Showing that the total height fallen is as the square of the total time. Again, during the successive seconds, the successive additional velocities acquired are : — time, I, I, I, I second, velocities acquired, 32.2, 32.2, 32.2, 32.2 feet per second And the total or final velocities acquired, reckoning from the commence- ment of the fall, are : — total times, i, 2, 3, 4 seconds, final velocities, 32.2, 32.2 x 2, 32.2 x 3, 32.2 x 4 feet per second. or 32.2, 64.4, 96.6, 128.8 feet per second. m Showing that the velocity acquired is in direct proportion to the time of the fall. The above relations of time, height, and velocity are brought together for comparison, thus : — time as, i, velocity acquired as, i, height of fall as, i, or as I, Showing that, whilst the velocity increases simply with the time, the lieight of fall increases as the square of the time, and as the square of the velocity. The force of gravity is expressed by the velocity communicated by gravity to a body falling freely in a second, namely, 32.2 feet per second, and is symbolized by g. The foregoing relations of time^ velocity^ and height of fall, are comprised in the six following propositions with their answers — applicable to the condition of a body falling freely. They are much used in mechanical calculations. I and 2. Given the time^ to find the velocity and the height, 3 and 4. Given the velocity ^ to find the time and the height, 5 and 6. Given the height, to find the time and the velocity. 2, Zy 4, &C. 2| 3» 4, &C 4, 9» 16, ^C. 2^ 6 y 42, &c. GRAVITY. 279 Rules for the Action of Gravity. Putting /=the time of falling in seconds, v = the velocity in feet per second, ^=the height of fall in feet, and ^= gravity or 32.2; then Rule i. Given the time of fall, to find the velocity acquired by a falling body. Multiply the time in seconds by 32.2, the product is the final velocity in feet per second. Or Z/ = 32.2 / ( I ) Rule 2. Given the time of fall, to find the height of the fall. Multiply the square of the time in seconds by 16.1. The product is the height of fall in feet. Or ^=16.1 /2 (2) Rule 3. Given the velocity^ to find the time of falling. Divide the velocity in feet per second by 32.2. The quotient is the time in seconds. Or /=-^ (3) 32.2 Rule 4. Given the velocity^ to find the height of fall "due" to the velocity. Square the velocity in feet per second, and divide by 64.4. The quotient is the height of fall in feet Or /i=g^ (4) 64.4 Rule 5. Given the height of fall, to find the time 6f falling. Divide the height in feet by 16.1, and find the square root of the quotient The root is the time in seconds. Or / or / = K\/ h (s) Rule 6. Given the height of fall, to find the velocity due to the height Multiply the height in feet by 64.4, and find the square root of the product The root is the velocity in feet per second. Or, multiply the square root of the height in feet by 8.025; the product is the velocity in feet per second. Note. — It is usual to take the integer 8 only for the multiplier. Sjonbolically, these operations are expressed as follows : — z/ = 32.2\/-^ — = a/64.4^ = 8.025 a/ ^ j^ 32.2 ^ ^ OT in a round number z^= 8 a/ h '. (6) The above rules are applicable, inversely, to the motion of bodies pro^ jected upwards and uniformly retarded by gravity. The height to which a body projected vertically upwards by an initial impulse, will ascend, is equal to tie height through which the body must fall in order to acquire the initial velocity, and the same rule (4) applies in these two cases. 28o FUNDAMENTAL MECHANICAL PRINCIPLES. The following table, No. loi, gives the velocity acquired by a falling body in falling freely through a given height Table No. 102 gives, conversely, the height of fall due to a given velocity. Table No. 103 gives the fall and the final velocity due to a given time of falling freely. Table No. loi. — ^Velocity acquired by Falling Bodies, due to Given Heights of Fall. r^= 8.025 >y/ ^- Hdghtof Van Vdodty in Feet per Second. Height of FalL Velocity in Feet per Second. Height of FalL Velocity in Feet per Second. Hdgfatof Vdodty in Feet per Second. feet. ft. per sec. feet ft. per sec feet. ft. per sec feeL ft. per sec .01 .803 3-0 13.90 23 38.49 50 56.74 .02 1. 14 3-5 15.01 24 39.31 100 80.25 .03 1-39 4.0 16.05 25 40.12 150 98.28 .04 1.61 4.5 1703 26 40.92 200 "35 .05 1.80 5-0 17.99 27 41.70 300 139.0 .06 1.97 5-5 18.82 28 42.47 400 160.5 .07 2.12 6.0 19.66 29 43.22 500 179.9 .08 2.27 6.5 20.46 30 43.95 600 196.6 .09 2.41 7.0 21.23 31 44.68 700 212.3 .1 2.54 7.5 21.97 32 45.39 800 226.9 .2 3.20 8.0 22.69 33 46.10 900 240.7 •3 4.40 8.5 23.40 34 46.79 1000 253-8 .4 5'07 9.0 24.07 35 47.47 1500 310.8 •5 5.68 9-5 2473 36 48.15 2000 3589 .6 6.22 10 25.38 37 48.81 2500 401.2 .7 6.71 II 26.62 38 49.47 3000 439-5 .8 7.18 12 27.80 39 50.11 3500 474-7 •9 7.61 13 28.93 40 50.75 4000 507.S I.O 8.03 14 30-03 41 51.38 4500 538.3 1.2 8.79 15 31.08 42 52.01 5000 567-4 L4 9.50 16 32.10 43 52.62 6000 621.6 1.6 10.15 17 33.09 44 53.23 7000 671.4 1.8 10.77 18 34.05 45 53.83 8000 717.8 2.0 "35 19 34.98 46 54-43 9000 761.3 2.25 12.04 20 35.89 47 55.02 1 0000 802.5 2.50 12.69 21 36.77 48 55.60 2.75 13-31 22 37.64 49 56.17 GRAVITY. 281 Table No. 102. — Height of Fall due to Given Velocities. h^ 64.4" Velocity in Feet Height of Fall. Velocity in Feet Height of Fall Velocity in Feet Height of Fall Velocity in Feet Height of Fall. per Second. per Second. per Second. per Second. fc per sec feet. (t per sec. feet ft per sec feet ft. per sec feet •25 .0010 19 5.61 46 32.9 73 82.7 •50 .0039 20 6.21 47 34.3 74 85.0 ■75 .0087 21 6.85 48 35.8 75 87.4 1. 00 .016 22 7.52 49 37.3 80 99.4 1-25 .024 23 8.21 50 38.8 85 II2.2 1-50 .035 24 8.94 51 40.4 90 125.8 1-75 .048 25 9-71 52 42.0 95 140. 1 2 .062 26 10.5 53 43-6 JOG 155-3 2-5 .097 27 II-3 54 45-3 105 171. 2 3 .140 28 II. 2 55 47.0 no 187.9 3-5 .190 29 131 • 56 48.7 115 205.4 4 .248 30 14.0 57 50-4 120 223.6 4-5 .314 31 14.9 58 52.2 130 262.4 5 .388 32 159 59 54.1 140 304.3 6 .559 33 16.9 60 55-9 150 349-4 7 .761 34 17.9 61 57.8 175 475-5 8 .994 35 19.0 62 59.7 200 621 9 1.26 36 20.1 63 61.6 300 1397 JO 1.55 37 21.3 64 63.6 400 2484 XI 1.88 38 22.4 65 65.6 500 3882 J2 2.24 39 23.6 66 67.6 600 5590 13 2.62 40 24.9 67 69.7 700 7609 J4 304 41 26.1 68 71.8 800 9938 15 3-49 42 27.4 69 73-9 900 12578 16 398 43 28.7 70 76.1 1000 15528 17 4.49 44 30.1 71 78.3 '* 1 5'"^^ 45 31-4 72 80.5 282 FUNDAMENTAL MECHANICAL PRINCIPLES. Table No. 103. — Height of Fall and Velocity acquired, for Given Time of Fall. ^ = 16.1 A' V-^2,2t ' Velocity Velocity Velocity Time of Height of acquired in Feet per Time of Height of acquired in Feet per Time of Height of acquired in Feet per Fall Fall. Fall FalL Fall. Fall. Second. Second. Second. seconds. feet. ft. per sec. seconds. feet. ft. per sec. seconds. feet. ft. per sec I 16. 1 32.2 12 2318 386.4 23 8517 740.6 2 64.4 64.4 13 2721 418.6 24 9273 772.8 3 144.9 96.6 14 3156 450.8 25 10062 805.0 4 257.6 128.8 15 3623 483.0 26 10884 837.2 5 402.5 161. 16 4122 515-2 27 II737 869.4 6 579-6 193.2 17 4653 547.4 28 12622 901.6 7 788.9 225.4 18 5217 579-6 29 13540 933-8 8 1030 257.6 19 5812 611.8 30 14490 966.0 9 1304 289.8 20 6440 644.0 31 15473 998.2 10 1610 322.0 21 7100 676.2 32 16487 1030 II 1948 354.2 22 7792 708.4 ACCELERATED AND RETARDED MOTION IN GENERAL. The same rules and formulas that have been applied to the action of gravity are applicable to the action of any other uniformly accelerating force on a body, the numerical constants being adapted to the force. If an accelerating or retarding force be greater or less than gravity; that is to say, than the weight of the body, the constants 16.1, 32.2, and 64.4 are to be varied in the same proportion. To do this, multiply the constant by the accelerating force, and divide the product by the weight of the body. Let / be the accelerating force, and w the weight of the body, then the constant becomes 16.1/ ^^ 32.2/ ^^ 64.4/. or or w w w (a) and substituting this in the formulas (i) to (6) for gravity, the following general rules and formulas are arrived at for the action of uniformly accel- erating or retarding forces. The rules are written for accelerating forces, but they apply by simple inversion to retarding forces also. General Rules for Accelerating Forces. The accelerating force and the weight of the body are expressed in the same unit of weight; and the space in feet, the time in seconds, and the velocity in feet per second. In the following rules the time during which a body is acted on by an accelerating force is called tAe titne; the velocity acquired at the end of the ACCELERATED AND RETARDED MOTION. 283 time is called the final velocity; the space traversed by the body during the time is called the space; the accelerating force is called the force, t - the time. V = the final velocity. s = the space. / = the force. w = the weight. Rule 7. Given the weig/it^ the force^ and the time; to find the final velocity. Multiply the force by the time and by 32.2, and divide by the weight The quotient is the final velocity. Or Rule 8. Given the weighty the forccy and the time; to find the space. Multiply the force by the square of the time and by 16. i, and divide by the weight Or . = 1^:11^ (8) Rule 9. Given the weight, the filial velocity, and the force; to find the time. Multiply the final velocity by the weight, and divide by the force, and by 32.2. The quotient is the time. Or . W V / X 32.2/ Rule 10. Given the weight, \ht final velocity, and the force; to find the space. Miiltiply the weight by the square of the velocity, and divide by the force, and by 64.4. The quotient is the space. Or W 7/^ / V '=-e^f ('°> Rule ii. Given the weight, the force, and the space; to find the time. Multiply the weight by the space, and divide by the force; find the square root of the quotient, and divide it by 4. The last quotient is the time in seconds. Or ^-Hyf^ (") Rule i 2. Given the weight, the force, and the space; to find the final velocity. Multiply the space by the force, and divide by the weight; find the square root of the quotient, and multiply by 8. The product is the final velocity. Or '7^ V=%J^- (12) w ' Rule 13. Given the weight, the space, and \he final velocity; to find the force. Multiply the weight by the square of the final velocity, and divide by the space, and by 64.4. The quotient is the force. Or ^=6iT* <'3) 284 FUNDAMENTAL MECHANICAL PRINCIPLES. Rule 14. Given the weigJit^ timcy and final velocity; to find the force. Multiply the weight by the square of the velocity, and divide by the space, and by 32.2. Or /=^, (X4) 32.2 / Note I. When the accelerating or retarding force bears a simple ratio to the weight of the body, the ratio may, for greater readiness in calculation, be substituted in the quantities (a) representing the modified constants, for the force and the weight. Suppose the accelerating force is a tenth part of the weight, then the ratio is i to 10, and 16. 1 , __ = 1. 61, 10 ^2.2 ^ =3.22, 10 644 _ 10 6.44; and these quotients may be substituted for 16. i, 32.2, and 64.4 respectively in the formulas for the action of gravity (i) to (6), to fit them for direct use in dealing with an accelerating force one-tenth of gravity, the height h in those formulas, of course, being taken to mean space traversed. Note 2. The tables, Nos. 101-103, pages 280-282, for the relations of the velocity and height of falling bodies, may be employed in solving questions of accelerating force generally. Example, A ball weighing 10 lbs. is projected with an initial velocity of 60 feet per second on a level bowling-green, and the frictional resistance to its motion over the green is i lb. . What distance will it traverse before it comes to a state of rest? By rule 10, 10 lbs. X 60^ ^^^ .. — Tu z — = 559 leet, I lb. X 64.4 the distance traversed. Again, th^ same result may be arrived at, according to Note i, by- multiplying the constant 64.4, in rule 4, for gravity, by the ratio of the force and the weight, which in this case is -j^, and 64.4 x -j^ = 6.44. Substituting 6.44 for 64.4 in that rule and formula, the formula becomes Tj 60 ^ = T — ^-f- — = 559 feet, 6.44 6.44 the distance traversed, as already found. But the question may be answered more directly by the aid of the table for falling bodies (No. 102, page 281). The height due to a velocity of 60 feet per second, is 55.9 feet; and it is to be multiplied by the inverse ratio of the accelerating force and the weight of the body, or ^, or 10; that is, 55.9 X 10 = 559 feet, the distance traversed. If the question be put otherwise — What space will a ball move over before it comes to a state of rest, with an initial velocity of 60 feet per GRAVITATION ON INCLINED PLANES. 285 second, allowing the friction to be i-ioth the weight of the ball? The answer may be given, that the friction, which is the retarding force, being i-ioth of the weight, that is of gravity, the space described will be 10 times as much as is necessary for gravity, supposing the ball to be projected vertically upwards to bring the ball to a state of rest. The height due to the velocity is 55.9 feet; then 55.9 X 10 = 559 feet, the space described by the ball. Action of Gravity on Inclined Planes. If a body freely descend an inclined plane by the force of gravity alone, the velocity acquired by the body when it arrives at the foot of the plane, is that which it would acquire by falling freely through the vertical height. Or, the velocity is that " due " to the height of the plane. The time occupied in making the descent is greater than that due to the height, in the ratio of the length of the plane, or distance travelled, to the height. The time is therefore directly in proportion to the length of rhe plajie, when the height is the same. The impelling or accelerating force by gravitation acting in a direction parallel to the plane, is less than the weight of the body, in the ratio of the height of the plane to its length. It is, therefore, inversely in proportion to the length of the plane, when the height is the same. The time of descent, under these conditions, is inversely in proportion to the accelerating force. If, for instance, the length of the plane be five times the height, the time of making freely the descent on the plane by gravitation is five times that in which a body would freely fall vertically through the height; and the impelling force down the plane is '/j th of the weight of the body. Problems on the descent of bodies on inclined planes are soluble by the aid of the rules 7 to 14, for the relations of accelerating forces. But, as a preliminary step, the accelerating force is to be determined, by multiplying the weight of the descending body by the height of the plane, and dividing the product by the length of the plane. For example, let a body of 15 pounds weight gravitate freely down an inclined plane, the length of which is fiver times the height, the accelerating force is 1 5 -r 5 = 3 pounds. If the length of the plane be 100 feet, the height is 20 feet, and the velocity acquired in falling freely firom the top to the bottom of the plane would be, by rule 12, z/ = 8/y/5-^^ — 2z=s^ 20 =35.776 feet per second. The time occupied in making the descent is, by rule 11, ^=}i V i^-^^^^ = H\/ 500 = 5-59 seconds. ^ ' 3 Whereas, for a free vertical fall through the height, 20 feet, the time would be, by rule 5, t=%^ 20 = I.I 18 seconds, which is '/s ^ ^^ ^^ ^^^^ ^^ making the descent on the inclined plane. 286 FUNDAMENTAL MECHANICAL PRINCIPLES. Special Rules for the Descent on Inclined Planes. The height and the length of an inclined plane may be substituted for the accelerating force and the weight respectively in the rule (ii), to find the time. Putting ^ = the height bf the plane, and /= the length of the plane, the formula (ii) '= Vk V -^ becomes t=}iy/ ^=% \/-t-, / "^ V A '* V A / °'-''=i7=r (^5) Rule 15. — Given the length and the height of the inclined plane, to find the time in which a body would freely descend by gravitation. Divide the length by four times the square root of the height; the quotient is the time in seconds. For example, the length of the plane is 100 feet, and the height is 20 feet, and the time is 100 ^"47W =5-59 seconds, as was found before. Again, by inversion of the formula (15), = 4/\/ h , and then ^=1^ (^^) Rule 16. — Given the length of the inclined plane, and the time of free descent by gravitation, to find the height through which the body descends. Divide the square of the length by the square of the time in seconds and by 16; the quotient is the length of the inclined plane. For example, the length of the plane is 100 feet, and the time of descent is 5.59 seconds; then the vertical height of the descent is h = 5 = 20 feet, the height. 5.59^x16 Average Velocity of a Moving Body Uniformly Accelerated OR Retarded. The average velocity of a moving body uniformly accelerated or retarded, during a given time or in a given space, is equal to half the sum of the initial and final velocities; and if the body begin from a state of rest or arrive at a state of rest, the average speed is half the final or initial velocity, as the case may be. Thus, in the example of a ball rolling, the initial speed or velocity is, in either case, 60 feet per second, and the terminal speed is nothing; the average speed is therefore the same, namely, one-half of that, or 30 feet per second MASS. — CENTRE OF GRAVITY. 287 MASS. Weight IS not an essential property of a body; it is only the attraction of the earth exerted upon the body. Suppose the attractive force to be suspended, then the body would cease to have weight What would remain? Mass, or substance, simply. But, though weight is not mass, it is a direct measure of mass, in the same locality, or wherever the force of gravitation is the same, for double the mass has twice the weight Weight alone, however, is not sufficient as a universal measure of mass, since the weight of the same mass would vary according to the force of gravitation for different situations. The mass, therefore, varies inversely as the force of gravitation, when the weight remains the same. That force is measurable by the height through which a body falls in a given time, or by the velocity acquired at the end of that time, say, a second, expressed by g. In its most general form, then, the expression for the mass of a body comprises the weight directly and the force of gravitation inversely; thus ^ = -T (17) in which m is the mass, w the weight, g the force of gravitation; that is to say, the mass of. a body is equal to the weight of the body divided by the force of gravity. Since the weight and the force of gravity vary in the same ratio, the mass — of a body is the same at all places. As the quan- tity of matter of the same body is also constant whatever place it occupies, the mass — gives an exact idea of the quantity of matter, and is a measure of it MECHANICAL CENTRES. There are four mechanical centres of force in bodies, namely, the centre of gravity, the centre of gyration, the centre of oscillation, and the centre of percussion. Centre of Gravity. The centre of gravity is the physical centre of a body, or of a system of bodies in rigid connection with each other, about which the gravitation of the several particles of the body is self-balanced, and at which it can be freely suspended or supported in any position in a state of rest. In various calculations, the whole weight or mass of a body is considered as placed at its centre of gravity. To find the centre of gravity of any plane figure mechanically: — Suspend the figure by any point near its edge, and mark on it the direction of a plumb-line hung from that point; then suspend it from some other point, and again mark the direction of the plumb-line in like manner. Then the centre of gravity of the surface will be at the point of intersection of the two marks of the plumb-line. The centre of gravity of parallel-sided objects may readily be found in this way. For instance, to find the centre of gravity of the arch of a bridge; draw the elevation upon paper to a scale, cut out the figure, and proceed v/ith it as above directed, in order to find the position of the centre of 288 FUNDAMENTAL MECHANICAL PRINCIPLES. gravity in elevation for the model. In the actual arch, the centre of gravity will have the same relative position as in the paper model In regular figures or solids the centre of gravity is the same as their geometrical centres. Thus, the centre of gravity of a straight line, a parallelogram, a prism, a cylinder, a circle, the circumference of a circle, a ring, a sphere, and a regular polygon, is the geometrical centre of these figures and solids respectively. To find the centre of gravity of a triangle; draw a straight line from one of its angles to the middle of the opposite side; the centre of gravity will be in this line at a distance of two-thirds of its length from the angle. Or, draw a straight line from two of the angles to the middle of the opposite sides respectively; the point of intersection of the two lines will be the centre of gravity. For a trapezium, or irregular four-sided figure, draw the two diagonals, and find the centres of gravity of each of the four triangles thus formed ; then join each opposite pair of these centres of gravity. The joining lines will cut each other in the centre of gravity of the figure. For a cone and a pyramid, the centre of gravity is in the axis or centre line, at a distance of three-fourths of the length of the axis from the vertex, or one-fourth from the base. For an arc of a circle, the centre of gravity lies in the radius bisecting the arc, and the distance of it from the centre of the arc is found by multiplying the radius by the chord of the arc, and dividing by the length of the arc; the quotient is the distance of the centre of gravity from the centre of the circle. For a sector of a circle, the centre of gravity is two-thirds of the distance of that of an arc, from the centre of the circle. It is found independently by multiplying the radius by twice the chord of the arc, and dividing by three times the length of the arc ; the quotient is the distance of the centre of gravity from the centre of the circle. For a parabolic space, the centre of gravity is in the axis, or centre line, and its distance from the vertex is three-fifths of the centre line or axis. For a paraboloid, the centre of gravity is in the axis, at a distance from the vertex of two-thirds of the axis. For two bodies, fixed or suspended one at each end of a straight bar, the common centre of gravity is in the bar, at that point which divides the distance between their individual centres of gravity, in the inverse ratio of the weights respectively. For example, if two weights of 20 lbs. and 10 lbs. be suspended on a bar at a distance of 9 feet apart between their centres of gravity, the common centre of gravity will divide the distance in the ratio of i to 2, being 3 feet from the heavier weight, and 6 feet from the lighter. In this example, the weight of the bar is neglected; but it may be allowed for according to the following direction. For more than two bodies connected in one system, the common centre of gravity may.be found by finding, in the first place, the common centre of gravity of two of them, and then finding that of these two jointly with a third, and so on to the last body in the group. Centre of Gyration. — Radius of Gyration. — Moment of Inertia. The centre of gyration, or revolution, is that point in a revolving body, or system of bodies, at a certain distance from tlie axis of motion, in which the whole of the matter in revolution may, as an equivalent condition, be CENTRE OF GYRATION. 289 conceived to be concentrated, just as If a pound of platinum were substituted for a pound of revolving feathers, whilst the moment of inertia remains the same. The work accumulated in the body, of which the moment of inertia is a measure, remains in such a case the same, at the same angular velocity; and, as a necessary consequence, if the same accelerating force be applied to the body at the centre of gyration, as would actually be expended over the distributed matter of the body to communicate to it its angular velocity, the same angular velocity would be generated. The distance of the centre of gyration from the axis of motion is called the radius of g)rration ; and the moment of inertia is equal to the product of the square of the radius of gyration by the mass or weight of the body. The moment of inertia of a revolving body is found exactly by ascertain- ing the moments of inertia of every particle separately, and adding them together; or, approximately, by adding together the moments of the small parts arrived at by the subdivision of the body. Rule i. To find the moment of inertia of a revolving body. Divide the body into small parts of regular figure. Multiply the mass, or the weight, of each part by the square of the distance of its centre of gravity from the axis of revolution. The sum of the products is the moment of inertia of the body. Note. — ^The value of the moment of inertia obtained by this process will be more nearly exact, the smaller and more numerous the parts into which the body is divided. Rule 2. To find the length of the radius of gyration of a body about a given axis of revolution. Divide the moment of inertia of the body by its mass, or its weight, and find the square root of the quotient. The square root is the length of the radius of gyration; or '-/l- ; (') in which /// is the moment of inertia, and w is the weight of the body. Note, — When the parts into which the body is divided are equal, the radius of gyration may be determined by taking the mean of all the squares of the distances of the parts from the axis of revolution, and finding the square root of the mean square. The following are useful examples of the radius of gyration of bodies: — In a straight bar, or a thin rectangular plate, revolving about one of its ends, the radius of gyration is equal to the length of the rod, multiplied by ^/ j^, or 0.5775. In a straight bar, or a thin rectangular plate, revolving about its centre, the radius of gyration is equal to half the length, multiplied by y y^y or 0.5775. The general expression for the radius of g}'ration in a straight bar revolving on any point of its length, is in which a and b are the lengths of the two parts of the bar; that is to say, 19 290 FUNDAMENTAL MECHANICAL PRINCIPLES. divide the sum of the cubes of the. two parts by three times the length of the bar, and extract the square root of the quotient. The root thus found is equal to the radius of gyration. In a circular plate, a solid wheel of uniform thickness, or a solid cylinder of any length, revolving on its axis, the radius of gyration is equal to the radius of the object, multiplied by y J4, or 0.7071. In a plane ring, like the rim of a fly-wheel, revolving on its axis, the radius of gyration is approximately equal to the square root of half the sum of the squares of the inside and outside radius of the rim. In a thin circular plate, put in motion round one of its diameters, the radius of gyration is equal to half the radius of the circle. For the circumference of a circle, revolving about a diameter, the radius of gyration is equal to the radius multiplied by 0.7071. In the circumference of a circle revolving about its ovm. axis, the radius of gyration is equal to the radius of the circle. In a solid sphere revolving about a diameter, the radius of gyration is equal to the radius multiplied by V V5, or 0.6324. In the surface of a sphere, or an insensibly thin spherical shell, the radius of gyration is equal to the radius multiplied by ^/ ^, or 0.8615. In a cone revolving about its axis, the radius of gyration is equal to the radius multiplied by 0.1783. Centre of Oscillation. The centre of oscillation of a body vibrating about a fixed axis or centre of suspension, by the action of gravity, is that point in which, if, as an equivalent condition, the whole matter of the vibrating body were concen- trated, the body would continue to vibrate in the same time. It is the resultant point of the whole vibrating energy, or of the action of gravity in causing oscillation. As the particles of the body further from the centre of suspension have greater velocity of vibration than those nearer to it, it is apparent that the centre of oscillation is more distant than the centre of gravity is from the axis of suspension, but it is situated in the centre line which passes from the axis through the centre of gravity. It differs also from the centre of gyration in this, that whilst the motion of oscillation is produced by the gravity of the body, that of gyration is caused by some other force acting at one place only. The radius of oscillation, or the distance of the centre of oscillation from the axis of suspension, is a third proportional to the distance of the centre of gravity from the axis of suspension and the radius of gyration. Hence the following rule for finding the radius of oscillation : — CENTRE OF OSCILLATION. — THE PENDULUM. 29I Rule 3. To find the radius of oscillation in a body vibrating on an axis. Square the radius of gyration of the body, and divide by the distance of the centre of gravity from the axis of suspension. The quotient is the radius of oscillation. Or, T» J- r n ^- radius^ of gyration. / ^ v Radius of oscillation = -j-. -z f^ 7-— p :— ( 3 ; distance of centre of gravity from axis. If the axis of suspension be in the vertex or uppermost point of a plane figure, and the motion be edgewise, then. In a right line, or straight rod, the radius of oscillation is two-thirds of the length of the rod. In an isosceles, or equal-sided triangle, it is three-fourths of the height of the triangle. In a circle it is five-eighths of the diameter. In a parabola it is five-sevenths of the height. But, if the oscillation of the plane figure be sidewise, then, In a circle suspended at the circumference, the radius of oscillation is three-fourths of the diameter. In a rectangle suspended by one of its angles, it is two-thirds of the diagonal. In a parabola suspended by the vertex, it is five-sevenths of the axis plus one-third of the parameter. In a parabola suspended by the middle of its base, it is four-sevenths of the axis plus half the parameter. In a sector of a circle suspended by the centre, it is three-fourths of the radius multiplied by the length of the arc, and divided by the length of the chord. In a cone it is four-fifths of the axis, plus the quotient obtained by dividing the square of the radius of the base by five times the axis. In a sphere it is two-fifths of the square of the radius divided by the sum of the radius and the length of the cord by which the sphere is suspended, plus the radius and the length of the cord. For example, in a sphere 16 inches in diameter, suspended by a cord 25 inches long, the radius of oscillation is 2x8^ -h 8 -I- 25 = 0.78 + 33 = 33.78 inches. 5(8 + 25) or 0.78 inch below the centre of the sphere. It may be noted that the depression of the centre of oscillation below the centre of the sphere, namely, 0.78 inch, is signified in the first quantity in this equation. 77^ Pendulum. A "simple pendulum" is the most elementary form of oscillating body, — consisting theoretically of a heavy particle attached to one end of a cord, or an inflexible rod, without weight, and caused to vibrate on an axis at the other end, or the centre of suspension. If an ordinary pendulum be inverted, so that the centre of oscillation shall become the centre of suspension, then the first centre of suspension will become the new centre of oscillation, and the pendulum will vibrate in the 292 FUNDAMENTAL MECHANICAL PRINCIPLES. same time as before. This reciprocal action of the pendulum is a property of all pendulous bodies, and it is known as the reciprocity of the pendulum. The time of vibration of an ordinary pendulum depends on the angle or the arc of vibration, and is greater when the arc of vibration is greater, but in a very much smaller proportion; and if this arc do not exceed 4° or 5°, that is to say, from 2° to 2^° on each side of the vertical line, the time of vibration is sensibly the same, however the length of the arc may vary within that limit. This property of a pendulum, of equal times of vibration, is known as isochronism. To construct a pendulum such that the time of vibration shall be the same whatever the magnitude of the angle of vibration may be, it is neces- sary to cause the pendulum to vibrate, not in a circular arc, but in a cycloidal curve. For this object the pendulum is suspended by a flexible thread or rod, which oscillates between two cycloidal surfaces, on which it alternately laps and unlaps itself; these are generated by a circle of which the diameter is equal to half the length of the pendulum. By means of the circle o b. Fig. 98, for example, of which the diameter is half the length of the pendulum, describe the right and left cycloidal curves oca, go' a', on the horizontal line a a'; and draw the tangent c B c', touching the cycloids at the middle of their lengths. The half-lengths o c, o c', are equal to twice the diameter of the generating circle OB, and consequently equal to the length of the pendulum, which Fig. 98.— Cycloidal Pendulum. will vibrate in cqual times, on the centre of suspension o, between the entire half- lengths o c, o c', or in any shorter path. The curve c p c' thus described by the pendulum, is itself a cycloidal curve, and is a duplicate of the other cycloids. Though a cycloidal motion of the pendulum is necessary to render it isochronous for all angles of vibration, yet taking very small arcs of the cycloidal path on either side of the vertical line, they do not sensibly differ from the circular arcs which would be described by an ordinary pendulum of the same length (o p) swinging freely. Hence the reason that the ordinary I>endulum vibrates in equal times when its vibrations do not exceed 4° or 5° in extent. The length of the pendulum vibrating seconds at the level of the sea in the latitude of London is 39.1393 inches, nearly a metre; at Paris it is 39.1279; at Edinburgh it is 39.1555 inches; at New York, 39.10153 inches; at the equator it is 39.027 inches, and at the pole it is 39.197 inches. Generally, if the force of gravity, or the length of the seconds pendulum at the equator be represented by i, the gravity, or the length of pendulum at other latitudes will be as follows : — Length of Seconds Pendulum, At the equator i. 00000 „ 30'' latitude 1.00141 „ 45 » 1.00283 » 52 „ 1.00357 ,, 60 „ 1.00423 „ 90 „ (the pole) 1.00567 THE PENDULUM. 293 According to these ratios, the force of gravity, and the length of the seconds pendulum, at the pole, are Viyeth greater than at the equator; there being a difference of length of between a fourth and a fifth of an inch. The following are the relations of the lengths of pendulums and the times of their vibrations, that is to say, of such as vibrate through equal angles, or of which the total angle of vibration does not exceed 4° or 5°: — The times of vibration of pendulums are proportional to the square root of the lengths of the pendulums. Conversely, the lengths oT pendulums are to each other as the squares of the times of one vibration, or inversely as the squares of the numbers of vibrations in a given time. The length of the seconds pendulum at London, 39.1393 inches, may be taken as a datum for calculation applicable to pendulums of different lengths, and to different times of vibration. Rule 4. To find the time of vibration of a pendulum of a given length. Divide the square root of the given length in inches by the square root of 39.1393, or 6.2561. The quotient is the time of a vibration in seconds. Or ^ 391393 6.2561' in which / is the given length of pendulum in inches, and / the time of vibration in seconds. Rule 5. To find -the number of vibrations per second of a pendulum of given length. Divide 6.2561 by the square root of the length in inches. The quotient is the number of vibrations per second. For the number of vibrations per minute. Divide 375.366 by the square root of the length in inches. The quotient is the number of vibrations per minute. Or « = -^^^ (per second); ( 5 ) n = ^l^^ (^rmmute); (5) in which n is the number of vibrations. Rule 6. To find the length of a pendulum when the time of a vibration is given. Multiply the square of the time of one vibration in seconds by 39.1393. The product is the length of the pendulum in inches. Or /=/2x39.i393 (6) Rule 7. To find the length of a pendulum when the number of vibrations per second is given. Divide 39.1393 by the square of the num- ber of vibrations in a second. The quotient is the length of the pendulum in inches. When the number of vibrations per minute is given. Divide 140,900 by the square of the number of vibrations in a minute. The quotient is the length of the pendulum in inches. Or /, 39-1393 /.) «^ (per second)' /_ 140,900 (7) n^ (per minute) 294 FUNDAMENTAL MECHANICAL PRINCIFl.ES. A pendulum may be shorteneci and yet vibrate in the same time as before, by the action of a second weight fixed on the pendulum rod above the centre of suspension. Here the upper weight counteracts the lower, and there is only the balance of gravitating force due to the preponderance of the lower weight available for vibrating both masses. The mass being thus increased while the gravitating force is diminished, a longer time is required for each vibration when the length of pendulum remains unaltered, or the pendulum may be shortened so that the time of the vibrations con- tinues the same. By varying the height of the upper weight above, the centre of suspension, and thus varying the level of the common centre of gravity, the period of vibration is varied in proportion. Rule 8. To find the weight of the upper bob of a compound pendulum necessary to vibrate seconds, when the weight of the lower bob is given, and the respective distances of the bobs from the centre of suspension. Multiply the distance in inches of the lower bob from the centre of suspen- sion by 39.1393, and from the product subtract the square of that distance (i); again, multiply the distance in inches of the upper bob from the centre of suspension by 39.1393, and add the square of that distance (2); multiply the lower weight by the remainder (i), and divide by the sum (2). The quotient is the weight of the upper bob. Or ^ = w 99-i393xD)- D^ (8) (39-1393 >"^ + <^' in which D and d are the respective distances of the lower and upper bobs from the centre of suspension, and W, a/, their respective weights. Thus, by means of a second bob, pendulums of small dimensions may be made to vibrate as slowly as may be desired. The metronome, an instrument for marking the time of music, is constructed on this principle, the upper weight being slid and adjusted on a graduated rod to measure fast or slow movements. The Centre of Percussion. If a blow is struck by an oscillating or revolving body moving about a fixed centre, the percussive action is the same as if the whole mass of the body were concentrated at the centre of oscillation. That is to say, the centre of percussion is identical with the centre of oscillation, and its position is found by the same rules as for the centre of oscillation. If an external body is so struck that the mean line of resistance passes through the centre of percussion, then the whole force of percussion is transmitted directly to the external body; on the contrary, if the revolving body be struck at the centre of percussion, the motion of the revolving body will be absolutely destroyed, so that the body shall not incline either way, just as if every particle separately had been struck. CENTRAL FORCES. When a body revolves on an axis, every particle moves in a circle of revolution, but would, if freed, move off in»a straight line, forming a tangent to the circle. The force required to prevent the body or partiple flying from the centre is called cctiiripetal force, and the tendency to fly from the centre is centrifugal force. These forces are equal and opposite — examples of action and reaction — and are classed as central forces. CENTRAL FORCES. 295 Centrifugal force varies as the square of the speed of revolution. It varies as the radius of the circle of revolution. It varies as the mass or the weight of the revolving body. Let c be the centrifugal force, w the weight of the revolving body, r the radius of revolution or gyration, m the mass of the body = -, in which ^ = 32.2 or gravity; and v the linear or circumferential velocity; then m v^ w v^ c— - = r 32.2 r That is to say, the centrifugal force of a revolving body is equal to the weight of the body multiplied by the square of the linear velocity, divided by 32.2 times the radius of gyration. If the height due to the velocity be substituted for the velocity in the above equation, the height // being equal to - — , then 64.4 2WV^ 2 w h €=- = , 64.4 r r and c \ w \ \ 2 h \ K That is to say, the centrifugal force is to the weight of the body as twice the height due to- the velocity is to the radius of gyration. From the first equation the following rules for revolving bodies are deduced, for finding one of the four elements when the other threie are given: — namely, the centrifugal force, the radius of gyration, the linear velocity, and the weight. Rule i. For the centrifugal force. Multiply the weight by the square of the speed, and divide by 32.2 times the radius of gyration. The quotient is the centrifiigal force. Or c^ (i) 32.2 r Rule 2. For the linear velocity. Multiply the centrifugal force by the radius of gyration, and by 32.2, and divide by the weight; and find the square root of the quotient. The root is the velocity. Or /x2.2cr / V ^ w Rule 3. For the weight Multiply the centrifugal force by the radius of gyration, and by 32.2, and divide by the square of the velocity. The quotient is the weight. Or 32.2 c r / V '^ = ^-:jr- : ; (3) Rule 4. For the radius of gyration. Multiply the weight by the square of the velocity, and divide by the centrifugal force, and by 32.2. The quotient is the radius of gyration. Or w iP' r-= (4) 32.2 c 296 FUNDAMENTAL MECHANICAL PRINCIPLES. Note, — ^When the velocity is expressed as angular velocity, in revolutions per unit of time, it is to be reduced to linear or circumferential velocity by multiplying it by the radius of gyration and by 6.28; or f = 6.28 if r, in which 7/ is the angular velocity. By substitution and reduction in equation (i), the following equation in terms of the angular velocity is arrived at : — 0.8165 c=wr7/^, /j) from which is found '=5176i = '-"S«"-^' (6) That is to say, the centrifugal force is equal to the weight multiplied by the radius of gyration and by the square of the angular velocity, and by 1.225. MECHANICAL ELEMENTS. The function of mechanism is to receive, concentrate, diffuse, and apply power to overcome resistance. The combinations of mechanism are num- berless; but the primary elements are only two, namely, the lever and the inclined plane. By the lever, power is transmitted by circular or angular action; that is to say, by action about an axis; by the inclined plane, it is transmitted by rectilineal action. The principle of the lever is the basis of the pulley and the wheel and axle; that of the inclined plane is the basis of the wedge and the screw. For the present, frictional resistance and the weight of the mechanism are not considered; the terminal resistance is called the weight; and the elemental mechanisms are to be treated as in a state of equilibrium, in which the power exactly balances the weight without actual movement. The action, or work done, will be subsequently treated. The Lever. The elementary lever is an inflexible straight bar, turning on an axis or fixed point, called the fulcrum; the force being transmitted by angular motion about the fulcrum, from the point where the power is applied to the point where the weight is raised, or other resistance overcome. There are three varieties of the lever, according as the fulcrum, the weight, or the power is placed between the other two, but the action is, in every case, re- Pjg. 99-— Lever. duciblc to that of three parallel forces in equilibrium (page 275). First. The power is applied at one end ^, of the lever ab c. Fig. 99, and transmitted through the fulcrum, ^, to the weight at the other end c. The moments of the power and the weight about the fulcrum are equal, or power y^ab- weight y>b c. That is, the product of the power by its distance from the fulcrum is equal THE LEVER. 297 to the product of the weight by its distance from the fulctum. Conse- quently power : weight : : b c \ ab^ that is, the power and the weight are to each other inversely as their respective distances from the fulcrum. The ratio of the length of the power end of the lever to the length of the weight end is called the leverage of the power. The respective lengths, Fig. 99, being 7 feet and i foot, the leverage is 7 to i, or 7, The three varieties of the lever are grouped together in Figs. 100, 10 1, and 1 102. In each case, the lever is supposed I _, I to be 8 feet long and divided into feet. J ^ f \ -, — p 1 . ^. The leverage, in the first, is 7 to i, or 7; ^^ | "S in the second, 8 to i, or 8; in the third, 'f^ yi to I, or ^ : showing that, in the first Q I A case, the power balances seven times its yr ^ own amount; in the second case, eigh times its amount; in the third case, only Fig. xoo. — Lever, ist kind. ^ + R T ' ' -r- Ji a II'''' IT Fig. loi. — Lever, ad kind. Fig. loa. — Lever, 3d kind. one-eighth of itself, because it is nearer to the fulcrum than the weight. In each case the moments of the power and the weight about the fulcrum are equal, for, in each case. Pxdf ^ = Wx3^. {a) The pressures exerted at the extremities of the lever act in the same direction, and the sum of them is equal and opposite to the intermediate pressure, whether it be that of the fulcrum, the weight, or the power ( — ). From this the pressure on the fulcrum may be found. If it be in the middle, the pressure is equal to the sum of the power and the weight, that is, 60 + 420 = 480 lbs. in the example above ; if at one end, it is equal to the difference of them, that is, it is 480 — 60 = 420 lbs. when the weight is in the middle, and it is 60-7^^ = 52)^ lbs. when the power is in the middle. From the equation for the equality of moments, orPxL =Wx/, {b) in which L and / are the respective distances of the power and the weight from the fulcrum, rules may be formed for finding any one of the four quantities, when the other three are given. Rule i. To find the power. Multiply the weight by its distance from the fulcrum, and divide by the distance of the power from the fulcrum. The quotient is the power. 298 FUNDAMENTAL MECHANICAL PRINCIPLES. Or, divide the weight by the leverage. The quotient is the power. Or Rule 2. To find the weight Multiply the power by its distance firom the fulcrum, and divide by the distance of the weight from the fulcrum. The quotient is the weight. Or, multiply the power by the leverage. The product is the weight Or w=y- (2) Rule 3. To find the distance of the power from the fulcrum. Multiply the weight by its distance from the fulcrum, and divide by the power. The quotient is the distance of the power from the fulcrum. Or ^ = 17 (3> Rule 4. To find the distance of the weight from the fulcrum. Multiply the power by its distance from the fulcrum, and divide by the weight The quotient is the distance of the weight from the fulcrum. Or 9 If the weight of the lever be included in such calculations, its influence is the same as if its whole weight or its mass were collected at its centre of gravity. Thus, if the lever of the first kind, Fig. 100, weighs 30 lbs., and its centre of gravity be at the middle of its length, the weight of the lever co-operates with the power, at a mean distance of 3 feet from the fulcrum. By equality of moments (P X 7) X (30 X 3) = W X I = 420 lbs. X I, and P X 7 = 420 - 90 = 330 lbs.; therefore P, the power at the end of the lever required to balance the Fig. Z03. — Inclined Lever. Fig. xo4.^Inclined Lever. weight, is only 3304-7 = 47.1 lbs. in co-operation with the weight of the lever, as compared with 60 lbs., without reckoning the aid from this source. When the lever is inclined to the direction of the forces, as in Fig. 103, THE LEVER. 299 equilibrium, or the equality of moments, may nevertheless be maintained. Drawing the horizontal line a' b d through the fulcrum, to meet the ver- ticals through the power and the weight at a! and ^, the moments of the power and the weight are to be estimated on the horizontal lengths ci b, b c'\ and the moment V %a! b- the moment W xb i/. The equality of moments may be proved in another way. Let the power and the weight be resolved, in order to find the pressures on the ends of the lever, at right angles to it, and thus to arrive at the moments as estimated on the actual length of the lever. Let the verticals through the ends of the lever, a m and cn^ Fig. 104, represent the power and the weight respectively, and draw a Y and c W perpendicular to the lever, and /// P' and n W parallel to if, completing the triangles a m P', c n W. Then a P' and c W are the components of the power and the weight respectively tending to turn the lever; and, it may be added, they bear the same ratio to each other as the power and the weight Consequently, if these com- ponents be multiplied by the respective lengths of the lever, the products will be the moments of the components, and the moments will be equal; or the moment ^ P' x df ^ = the moment cW xb c. These two methods of analyzing and finding the moments of the forces acting on an inclined lever — one, combining a reduced length of lever witjj the whole power and weight; the other, combining the whole length of lever with a reduced power and weight — lead to one conclusion, that a lever, if balanced in one position, is balanced in every other position, when the forces continue to act in parallel lines. ■® O Fig. 105.— Bent Lever. Fig. X06. — Bent Lever. The conditions of equilibrium in a bent lever may be defined sinriilarly. Let the lever a b c. Fig. 105, be bent at the fulcrum b; draw the horizontal line of b d, then the moments of the power and the weight are reckoned on the lines a' b, b c*, and they are equal to each other; or 300 FUNDAMENTAL MECHANICAL PRINCIPLES. »' «• Again, let the forces acting on a lever, whether straight or bent, be otherwise than vertical or parallel. When the arms of the lever are at right angles, and the power and the weight applied at right angles to the arms, as in Fig. io6, the moments are reckoned directly on the arms, ab,bc^ as in a straight lever; and the moment P x « ^ ^ the moment W x ^ r. The thrust, or pressure on the fulcrum, is in this case less than the sum of the power and the weight; and it may be determined by constructing a parallelogram upon the two arms of the lever, the arms representing inversely the respective forces. That is, a b represents the magnitude and direction of the weight W, and b c th®se of the power P. The diagonal b y, of the parallelogram repre- sents the magnitude and direction of the third force acting at the fulcrum to oppose O^ ^ the other two and maintain equilibrium. When the same lever is tilted into an IT oblique position, the power continuing to Fig. 107.— Bent Lever. act horizontally on the lever, Fig. 107, draw the vertical b' d through the end c of the lever, and produce the power line ap \.o meet it at }/, Complete the parallelogram a! 1/ (f b; then the sides a* b and b d zi^ the perpendiculars to the directions to the power and weight, on which the moments are reckoned, so that the moment P x a' ^ = the moment W x ^ ^. The diagonal ^ ^ is the resultant force at the fulcrum. ^ ^ 6 Fig. X08. — Bent I.,ever. Fl?. T09. — Serpentine Lever. If the power do not act horizontally, but in some other direction, a /, Fig. 108, produce the power-line pa and draw ba^ perpendicular to it. THE LEVER. 3OI Draw b (f 2& before ; then the moments are reckoned on the perpendiculars b (fy b c\ and, as before, To find the resultant force at the fulcrum. On the fulcrum ^ as a centre describe arcs of circles with the radii b a' and b (f, and draw b a^y b (f respectively parallel to the directions of the weight and the power, to cut the arcs at (f and c'^. Complete the parallelogram, and the diagonal b b^ represents in magnitude and direction the resultant force at the fulcrum. In this solution the power and the weight are assumed to act exactly, or sensibly, in the same plane. Again, in the serpentine lever a b c^ Fig. 109, supposed to be a pump- handle, the power P is applied obliquely in the direction a P. Produce P a and W c, and draw tlie perpendiculars b af^b <f from the fulcrum for the lengths of the- moments, then Pxrt'^ = Wx^^. Construct the parallelogram, as in the foregoing figure, and the diagonal b b" represents the resultant force at the fulcrum. ^/ By similar treatment the action of the . :'" ^^^^^"^. forces in levers of the second and third : kinds may be analyzed. The lever of the ®'~'-^,"yf^^ Av second kind, a c by Fig. 1 10, in an oblique \ \^^^^ C-/ position, is acted on horizontally by the : * -P power and the weight at a and c; draw /'^ the vertical b d a!y then b c' and b of are \^ the distances at which the forces act from W the fulcrum, or the lengths of the mo- Fig. no.— Lever of the ad kind. ments, and and the horizontal resultant force at the fulcrum is equal to the difference of the weight and the power. If more than two forces be applied to a lever in a state of equilibrium, the sum of the moments tending to turn tlie lever in one direction is equal to the sum of those tending in the opposite direction. If two or more levers are connected consecutively one to the other, so that they act as one system, with the power and the weight at the extremi- ties, then, in equilibrium, the ratio of the power to the weight is the product of the separate inverse ratios of all the levers. For example, in a connected series of three levers, having each their arms in the ratio of 2 to i, the combined inverse ratio is found by multiplying 2 by 2 and by 2 ; thus first lever 2 to i ratio, second lever 2 to i ratio, third lever 2 to i ratio, compound ratio 8 to i. That is; the power is to the weight as i to 8. 302 FUNDAMENTAL MECHANICAL PRINCIPLES. The Pulley. The pulley is a wheel over which a cord, or chain, or band is passed, in order to transmit the force applied to the cord in another direction. It is equivalent to a continuous series of levers, with equal arms on one fulcrum or axis, and affords a continuous circular motion instead of the intermittent circular motion of one lever. The weight W, Fig. iii, is sustained by the power P, by means of a cord passed over the pulley A, in fixed supports, and the centre line abc represents the element of the lever, from the ends of which the power and the weight may be conceived to depend, turning on the fulcrum b. By equality of moments, V y.ab = Vf y.bc; and the arms or radii a b, b c being equaJ, the power is equal to the weight, and the counter-pressure at the fulcrum is equal to twice the weight. When the power and weight act in directions #t an angle with each other, as in Fig. 112, the acting radii ab, be, representing the element of a bent ^^V1lh^Vll»^^^vftf^^^1^l s>x-.v^ 7-vA.-*»a>vvvv •■ .,.^.-^^ •vr Fig. xxL—Pullcy. Fig. 112.— Pulley. Fig. 1x3. —Pulley. lever, are lines drawn from the centre perpendicular to the directions of the power and weight The power is equal to the weight, but the counter- pressure on the fulcrum is less than twice the weight, and is represented by the diagonal b V of the parallelogram formed by the radii bc^, bd, drawn from the fulcrum parallel to the directions of the power and the weight respec- tively. Another construction for the parallelogram of forces in the action of the pulley is obtained by producing the directions of the power and the weight beyond the pulley. Fig. 113, intersecting each other at y, then forming the parallelogram, and drawing the diagonal b' if 2& the resultant pressure on the fulcrum. Thus the single fixed pulley acts like a lever of the first kind, and simply changes the direction of force, without modifying the intensity of the power. But the pulley may be employed as a lever of the second kind by suspending the weight to the axis of the pulley, and fixing one end. of the cord to a point as a fulcrum point. Thus, in Fig. 114, the weight W is suspended from the axis c. Fig. 114. — Movable Pulley, as a lever of the 3d kind. THE PULLEY. $0$ the cord is fixed to the point i', and the power P acts through the diameter acd, in which ^ is the fulcrum. By equality of moments, that is, the product of the power by the diameter of the pulley is equal to the product of the weight by the radius of the pulley, and the leverage being as 2 to 1, the power is only half the weight. In acting as a lever of the third kind, the power is applied to the axis a. Fig. 115, one end of the cord b«ing fixed at ^, and the weight attached at the other end, c. In this case, by equality of moments the product of the power by Jhe radius of the pulley is equal to that of the weight by the diameter, and the leverage being as i to z, the power is twice the weight These demonstrations with respect to movable pulleys only apply to cases of parallel cords; that is to say, when the direction of the power is parallel to J,ie»trofihe dkini" that of the weight. If, on the contrary, they be inclined ' to each other, as in Fig. 116, in which the weight is suspended by the axis, the power becomes greater than half the weight, as is shown by the parallelogram of which the diagonal c'tf represents the weight, and the sides e'y, i (f, the pull on the fulcrum, and the power exerted to sustain the weight Each of these sides is greater than half the diagonal. Fig. ii«.-MovaMe PuBey. Fir. iij.-FullcyBladci. ComMrta/ions of PuUeys. — Fast and Loose Pulleys. — In these last two applications of the pulley, it becomes movable when in action, and by com- bining two or more movable pulleys on the same or different axles in one block, with one cord, the gain of power may be increased in the same pro- portion. The movable block A, Fig. 117, carrying the weight, is used 304 FUNDAMENTAL MECHANICAL PRINCIPLES, with a fixed counterpart E, the rope is attached by one end to the fixed block, and is passed over the movable and fixed pulleys, from one to Che other in succession, the power being applied to the other end, as at P. 'I'his system is known as &st and loose puliey-b locks. The fixed end of the rope is sometimes attached to the movable block. Rule i. To find the power necessary to balance a weight or resistance by means of a system of fast and loose pulleys. Divide the weight by the is carried; that is, the number of ropes which alock. The quotient is the power required to 2 rope is attached to the fixed block, the num- Ti the loose block is twice the number of mov- may be found by dividing the weight by twice :ys. • is attached to the movable block, the divisor imber of movable pulleys plus i. ber of movable pulleys; if the fixed end of the block, I: (.) ipe be attached to the movable block, ^ (..) ight or resistance that will be balanced by a system of fast and loose pulleys. Multiply the r by the number of ropes proceeding from the Lble block. The product is the required weight ■, when the rope is attached to the fixed block, ply the power by twice the number of movable ys. ■, when the rope is attached to the movable :, multiply the power by twice the number of Lble pulleys plus i. , in the first case, W-2»P; (2) t: second case, w.(»«+.)P M ;ain, a combination may be formed of a num- f movable pulleys, as in Fig. 118, each of which, C, is suspended by a cord, with one end fixed e roof and the other end fixed to the axis of ext pulley. The weight W is hung to the axis delivers half the weight to the second pulley B, veight hanging to it, or one-fourth of the first ey C; from which only one-eighth of the first de or neutral pulley D lo the power P. In THE PULLEY. 30s general the divisor for the power is 2", or the //th power of 2, n being the number of movable pulleys. Rule 3. To find the power necessary to balance a weight by means of a system of separate movable pulleys, with separate cords consecutively con- nected as above described. Divide the weight by that power of 2 of which the index is the number of movable pulleys. The cjuotient is the power or force required to balance the weight Or, divide and subdivide the weight successively by 2 as many times as there are movable pulleys to find the power required. Or P = W (3) Rule 4. To find the weight that can be balanced by a given power, by means of a system of separate movable pulleys as above described. Mul- tiply the power by that power of 2 of which the index is the number of movable pulleys. The product is the weight required. Or, multiply the power successively by 2 as many times as there are pulleys. Or W=PX2'' (4) Note. — It is necessary that the cords should be parallel to each other, as in the illustration, in order that the above rules, 3 and 4, may apply. Wheel and Axle. The wheel and axle may be likened to a couple of pulleys of different diameters united together on one axis, of which the larger, a, Fig. 1 19, is the wheel, and the smaller, c, the axle, with a common ful- crum, b', the centre line abc representing the elements of a lever. The weight W on the axle at c is balanced *by the power P, on the wheel at a. The moments are equal, or Pxfl!^ = Wx^r; and the power is to the weight inversely as their distances from the centre; or V :^Y :\ be : ab. Fig. 119.— Wheel ana Axle. If a crank handle be substituted for the wheel, making a windlass, the radius of the crank is substituted for that of the wheel in estimating the ratio of the forces. Of the four elements, namely, the radius of the wheel or crank, the radius of the axle or roller, the power, and the weight, if three be given, the fourth can be found as follows, putting R and r for the respective radii. Rule i. To find the power. Multiply the weight by the radius of the axle, and divide by the radius of the wheel. The quotient is the power. Or WxJ (i) Rule 2. To find the weight Multiply the power by the radius of the 20 3o6 FUNDAMENTAL MECHANICAL PRINCIPLES. wheel, and divide by the radius of the axle. The quotient is the weight Or W = Px — r (») Rule 3. To find the radius of the wheel. Multiply the weight by the radius of the axle, and divide by the power. The quotient is the radius of the wheel. Or R=^^ (3) Rule 4. To find the radius of the axle. Multiply the power by the radius of the wheel,* and divide by the weight. The quotient is the radius of the axle. Or PR / V (4) r- W Note, — The diameters of the wheel and the axle or roller may be employed in the calculations instead of the radii. Inclined Plane. The inclined plane is a slope, or a flat surface inclined to the horizon, on which weights may be raised. By such substitution of a sloping path for a direct vertical line of ascent, a given weight can be raised by a power which is less than the weight itself. There are three elements of calculation in the inclined plane: — the plane itself, A B, Fig. 120; the base, or horizontal length, AC; and the height or vertical rise B C ; together forming a right- angled triangle. The weight W is to be raised through a height equal to C B, and for that object is drawn up the slope from A to B. It is partly supported during the O^^^^v^ (^^)^^ ascent, and it is in virtue of this degree of ^'*"^^\i^7/ support given to the weight that such a ^dFC^ "dead pull" as that of a direct vertical lift is avoided, and that it can be raised by a power much less than its o\mi weight. Let the weight W be kept at rest on the incline by the power P, acting in the line b P', parallel to the plane. Draw the vertical line ^^ to represent the weight; also bV perpendicular to the plane, and complete the parallelogram V c. Then the vertical weight ba is equivalent to b b\ which is the measure of support given by the plane to the weight, and b c, which is the force of gravity tending to draw the weight down the plane. The power required to maintain the weight in equilibrium is represented by this force be. Thus, the power and the weight are in the ratio of beta b a. Since the triangle of forces abe is similar to the triangle of the incline A B C, the latter may be substituted for the former in determining the relative magnitude of the forces, and P : W :: ^^ : fl^ :: BC : AB, Fig. X30.--Incluied Plane. THE INCLINED PLANE. 307 that is, the power, acting parallel to the inclined plane, is to the weight, as the height of the plane to its length. Then, by equality of moments, PxAB = WxBC, or P X length of inclined plane = W x height of inclined plane (a) For example, take the length of the inclined plane, 24 feet; the height, 2 feet; and the weight to be raised, 360 lbs. The power required to balance the weight is equal to 360 x 2 h- 24 = 30 lbs. Again, the base, A C, of the inclined plane, represents the element of the pressure of the weight on the inclined plane. It is thus seen that the sides of the triangle formed by an inclined plane, its base, and its height, are respectively proportional as follows : — The inclined plane to the weight at rest on the plane. The base to the pressure of the weight on the plane. The height to the power acting parallel to the plane. When the power acts in a direction parallel to the base, as in Fig. 121, in which the power P supports the weight W in the direction d V\ parallel to the base; draw the vertical da to represent the weight, and the line d^ perpen- dicular to the incline, and complete the parallelogram d' c. The weight If a, de- composed, is equivalent to ^ ^', the per- pendicular to the incline, representing the pressure of the weight upon the plane, and d c, the force of traction, or the power P. Here the pressure ^ ^ on the plane is greater than the v/eight, and the power ^ ^ is greater than when the line of traction is parallel to the incline. The triangles adc, ABC, being similar, the ratios of the power and the weight are as follows : — F :W :: dc : ad :: BC : AC; {3) that is, they are to each other as the height of the plane to its base; and the inclined plane, the base, and the height, are respectively proportional as follows: — The inclined plane to the pressure of the weight on the plane. The base to the weight at rest on the plane. The height to the power acting parallel to- the base. If the power be applied in any direction above that which is parallel to the incline, though the pressure of the weight on the plane will be less than the weight itself, yet, as in the previous case, the power is greater than is necessary when it acts in a direction parallel to the plane. Thus, in Fig. 122, in which the power P acts at a divergent angle in the direction d P', draw the vertical da, the perpendicular dd\ to the plane, and the extension of the power line to c, and complete the parallelogram. Then, the weight is represented by d a, the pressure on the incline by d V, and the power by a ^ or ^ ^. Fig. lax. — Inclined Plane. 3o8 FUNDAMENTAL MECHANICAL PRINCIPLES. Fig. laa.— Inclined Plane. For comparison, the parallelogram that would represent the relative forces arising from a power acting parallel to the plane, is added on the figure in dotted lines extending to the angles b" and (f. It shows that the pressure on the plane is greater than when the power is di- vergent, but that the power is less. It follows that the longer the inclined plane, when the height is the same, the less is the power required to balance the weight; in fact, the power simply varies in the inverse ratio of the length of the plane. If two inclines, A B and B D, of unequal lengths and the same height, be united back to back on the line BC, then two weights, W and W, on the respective planes, connected by a cord over a pulley at the summit B, will balance each other, when they are in the ratio of the lengths of the planes on which they rest That is, W : W : : A B : B D. From the formula ( a ), rules may be formed for finding one of the following four elements when the other three are given, namely, the length of the inclined plane, the height of it, the weight, and the power to balance the weight when acting in a direction parallel to the incline. Rule i. To find the power. Multiply the weight by the height of the plane, and divide by the length. The quotient is the power. Rule 2. To find the weight Multiply the power by the length of the plane, and divide by the height. The quotient is the weight Rule 3. To find the height of the inclined plane. Multiply the power by the length, and divide by the weight The quotient is the height Rule 4. To find the length of the inclined plane. Multiply the weight by the height of the plane> and divide by the power. The quotient is the length. Identity of the Inclined Plane and the Lever, Though the inclined plane is distinguished from the lever in the mode of operation, inasmuch as there is no motion about a mechanical centre, as in the lever, yet the conditions of equilibrium on the inclined plane may be established on the principle of the lever. Suppose a round weight W kept at rest on the incline A B by a power P parallel to the incline, passing Fig. 123. — Double Inclined Plane. LEVERAGE ON THE INCLINED PLANE. 309 through the centre a. Draw ab perpendicular to the incline; the point b is the point of contact of the weight with the incline. Draw the vertical line a d, and the perpendicular ^ ^ to it. Then the lines ab, be form a bent lever a be, of which b is the fulcrum, and ab,be the arms. The weight acts at the extremity e of the short arm, and the power at the extremity a of the long arm; and the power and the weight are to each other inversely as the relative arms of the lever, ab,be. Now, as abe and A B C are similar triangles, the arms a b, be axe to each other as the length and the height A B, B C, of the incline, and 'P : W : : b e : a b : : B C : A B: Fig. 124. — Leverage ofh an Inclined Plane*. that is, the power is to the weight as the height of length, which is the proportion already established The ratio of the length of an inclined plane to the leverage of the plane, and the products of the the plane, and of the weight into the height of the moments of the power and the weight. Suppose, again, that the power is applied at P, a P, passed round and over the weight parallel the inclined plane to its ( « ) page 307). its height may be called power into the length of plane, may represent the Fig. 125, through a cord to the incline; then the C A Fig. 125. — Leverage on an Inclined Plane. Fig. I a6.— Wedge. diameter of the weight a b becomes the long arm of the lever a be, through which the power acts, being double the length of the arm a b, Fig. 1 24, where the power is applied at the centre of the weight. By thus doubling the leverage, the power is halved, and the ratio of the power to the weight is as half the height of the plane to its length. In this case there is the action of a movable pulley combined with an inclined plane; the rolling weight moved by a cord lapped round it, repre- senting a movable pulley with the weight attached to the axle. Thus the leverage of the power on the inclined plane can be doubled. The Wedge. The wedge is a pair of inclined planes united by their bases, or " back to back," as A B C B , Fig. 1 26. Whereas inclined planes are fixed, wedges are moved, and in the direction of the centre line C A, against a resistance equally acted on by both planes of the wedge. The function of the wedge 310 FUNDAMENTAL MECHANICAL PRINCIPLES. is to separate two bodies by force, or divide into two a single body. In some cases the w^edge is moved by blows, as in splitting timber; in others it is moved by pressure. The action by simple pressure is now to be con- sidered. The pressure P is applied to a wedge at the head B B' at right angles to the surface, and the resistance or "weight" to be overcome is opposed to the wedge and acts at right angles to the faces A B, A B', at the middle points of which, a, a, it is supposed, to be concentrated. Whilst the wedge and the power move through a space equal to the length of the wedge A C, the weight is moved or overcome through a space equal to the breadth of the wedge B B'; and, as the power is to the weight inversely as the spaces described, they are to each other directly as the breadth to the length of the wedge. That is, P : W : : B B' : A C, and the product of the power by the length of the wedge is equal to the product of the weight by the breadth of the wedge; or PxAC = WxBB; or P X length = W X breadth of wedge {c) By the aid of the parallelogram the same conclusions are arrived at Thus, in Fig. 126, produce the directions of the two resistances, W«, \N a, to meet in the middle of the wedge at by complete the parallelogram, and draw the diagonals aca and bb\ The diagonal b b' is the resultant of the two forces ab,ab, and represents the pressure on the head of the wedge. Again, in the triangle a be, whilst a b represents, in magnitude and direction, the perpendicular pressure of the resistance on the wedge, a c, which is perpen- dicular to the centre line of the wedge, represents, in magnitude and direction, the force applied in overcoming the resistance. The ratio of the power to the weight is therefore as bb' to a c. And, as the triangle abb' is similar to the triangle ABB', P : W :: ^^ : ^z^ :: BB' : AC; that is, the power is to the weight as the breadth of the wedge to its length. From the formula ( c ), the following rules for wedges acting under pres- sure, as distinct from impact, are deduced : — Rule i. To find the weight or transverse resistance. Multiply the power by the length of the wedge, and divide by the breadth of the head. The quotient is the weight. Rule 2. To find the power. Multiply the weight or transverse resistance by the breadth of the head, and divide by the length of the wedge. The quotient is the power. Rule 3. To find the length of the wedge. Multiply the weight by the breadth of the wedge, and divide by the power. The quotient is the length of the wedge. Rule 4. To find the breadth of the wedge. Multiply the power by the length of the wedge, and divide by the weight. The quotient is the breadth of the wedge. Note, — I. The length of the wedge is taken as the distance from the head to the point of intersection of the sides. THE SCREW. 311 2. The power may be applied at the point of the wedge by drawing, instead of at the head by pressing. 3. The power may be applied in a direction parallel to one of the sides of the wedge, and the relation of the power to the weight may be found by construction, in the same manner as for the inclined plane, when the power is applied in a direction parallel to the base. See proportion ( ^ ), page 307. The Screw. The screw is an inclined plane lapped round a cylinder. Take, for example, an inclined plane ABC, Fig. 127, and bend it into a circular form, resting on its base, Fig. 128, so that the ends meet The incline may be Fig. 127. continued winding upwards round the same axis, and thus winding or helical inclined planes of any required length and height may be con- structed. The helix thus arrived at being placed upon a solid cylinder, and the dead parts of the helix removed, the product is an ordinary screw. The inclined fillet is the " thread " of the screw, and the screw is called "external." But the thread may also be applied within a hollow cylinder, and then it is " internal," such as an ordinary " nut " is. The distance of two consecutive coils apart, measured from centre to centre, or from upper side to upper side, — ^literally the height of the inclined plane, — ^for one revolution, is Qie "pitch" of the screw. The effect of a screw is estimated in terms of the pitch and the radius of the handle employed to turn either it or the nut, one on the other; and the leverage of the power is the ratio of the circum- ference of the circle described by the power end of the handle to the pitch. The radius is to be measured to the central point where the power is applied. The circumference being equal to the radius multiplied by twice 3. 14 16, or 6.28, Fig. ia8. P : W :: / : /-X6.28, in which / is the pitch and r the radius; also 6.28 Pr=Wx/; {d) that is, 6.28 times the product of the power by the radius of the handle is equal to the product of the weight by the pitch. Whence the following rules relative to the power of a screw, for finding any one of those four quantities when the other three are given : — Rule i. To find the power. Multiply the weight by the pitch, and 312 FUNDAMENTAL MECHANICAL PRINCIPLES. divide by the radius of the handle and by 6.28. The quotient is the power. Or P = ^^ (X) 6.28 r ^^ Rule 2. To find the weight. Multiply the power by the radius and by 6.28, and divide by the pitch. The quotient is the weight. Or ^^6,28^r ^^j Rule 3. To find the pitch. Multiply the power by the radius of the handle and by 6.28, and divide by the weight. The quotient is the pitch. Oi ^ 6.28 Pr , V /=— w~ : ^^) Rule 4. To find the radial length of the handle. Multiply the weight by the pitch, and divide by the power and by 6.28. The quotient is the length of the handle. Or "j5^p- <^) JYofe. — When the power is applied through a wheel fixed to the screw, the acting diameter of the wheel may be substituted for the radius in the above rules and formulas, and the constant becomes 3.14. Similarly, should the power-wheel be fixed to the nut so as to turn it upon the screw, instead of the screw within the nut, the same sub- stitutions may be made. WORK. Work consists of the sustained exertion of pressure through space. The English unit of work is one foot-pound; that is, a pressure of one pound exerted through a space of one foot. The French unit of work is one kilogrammetre; that is, a pressure of one kilogramme exerted through a space of one metre. One kilogrammetre is equal to 7.233 foot-pounds. In the performance of work by means of mechanism, the work done upon the weight is equal to the work done by the power. This prin- ciple of the equality of work is deducible from the principle of the equality of moments, and is expressed generally by the equation PxH = Wx>4, (a) in which H is the height or space moved through by the power, and // the height or space moved through by the weight at the same time. It signifies that the product of the power by the space through which it has acted is equal to the product of the weight by the space through which it has acted. Again, P : W : : /i : H, signifying that the power is to the weight inversely as the respective heights or spaces moved through by them in the same time. WORK. — WORK WITH THE MECHANICAL ELEMENTS. 313 Work done with the Lever. On the principle of the equality of moments, the power and the weight in the lever, neglecting frictional resistance, are to each other inversely as the lengths of the arms upon which they act, that is, of their radii of motion; and inversely as the arcs or spaces passed through or described by the ends of the arms. If the weighted lever, Fig. 99, page 296, be moved by the power, the power descends through an arc at a, and the weight is raised through an arc at c. These arcs may be taken as the heights moved through, and are proportional to the lengths of the respective arms, ab^b c. In this example, these are as 7 to i, and if the power descend 7 inches the weight is raised only i inch; but the weight raised is seven times the power applied, and "what is gained in power is lost in speed," or, more correctly, in space moved through. The equality of work thus developed from the equality of moments is thus expressed power X arc a — weight x arc c («) To show this arithmetically, let the weight be raised through i foot; then, with a leverage of 7 to i, the power descends 7 feet, and taking it, as before, at 60 lbs., the weight it raises will be 60 lbs. x 7 = 420 lbs., and the equation of work is 60 lbs. X 7 feet = 420 lbs. x i foot, (or 420 foot-pounds) (or 420 foot-pounds). Again, power : weight : : arc c : arc a^ expressing the principle of virtual velocities, the relative velocities being indicated by the arcs «, c. Work done with the Pulley. In using the single fixed pulley. Fig. iii, page 302, the power is equal to the weight, and the spaces through which they move in the same time are equal. With the movable pulley, Fig. 114, the weight is suspended at the axle, and in raising the weight i foot, the power at the circumference, with a leverage of 2, passes through 2 feet and is only half the weight If P and W be 20 lbs. and 40 lbs. respectively, the equality of work is thus expressed — (P) 20 lbs. X 2 feet = (W) 40 lbs. x i foot = 40 foot-pounds; and by means of this pulley a weight double the power is raised half the height through which the power is applied. Conversely, when the weight is suspended at the circumference of the movable pulley, Fig. 115, and the power applied at the axle, the leverage is J^ ; the power is therefore double the weight, and moves through i foot whilst the weight moves through 2 feet. Thus (P) 40 lbs. X I foot = (W) 20 lbs. X 2 feet = 40 foot-pounds. In a system of fast and loose pulley blocks, Fig. 117, page 303, the power being equal to the weight divided by the number of ropes, then, by 314 FUNDAMENTAL MECHANICAL PRINCIPLES. equality of work, the space through which the power is moved is equal to the height through which the weight is raised, multiplied by the number of ropes. Suppose that there are three movable pulleys and six ropes; if the weight, izo lbs., be raised i foot) each rope is shortened i foot and the power is moved 6 feet And (P) 20 lbs. X 6 feet = (W) 120 lbs. x i foot = 120 foot-pounds. Work done with the Wheel and Axle. 'heel, Fig. iig, page 305, makes one revolution, the axle also The power descends or traverses a space equal to the cir- thewheel = 2 (ai) x 3.i4i6,whilst the weight is raised through to the circumference of the axle = 2 (i^:) x 3.1416. If the wheel be r foot 6 inches, and that of the axle 3 inches, the s are 9.42 feet and 1.57 feet, being as 6 to i; and the power t, conversely, are as 1 to 6. If the power be 20 lbs., then ;P) 20 lbs. X g.42 feet = {W) rao lbs. x 1.57 feet. (188.4 foot-pounds) (188.4 foot-pounds). Work done with the Inclined Plane. is raised in opposition to gravity, and the work done on it is the product of the weight into the vertical height of the ;. The work done by the power is enpressed by the product into the length of the'plane, These two products express es of work, and Px/=Wx^, nated at (a), page 307, to express equality of moments. e, the length of the plane is 24 feet and the height 2 feetj 1,20 lbs., the power 10 lbs. Then, the work done in rising the whole of the incline is 240 lbs., thus (P) 10 lbs. X 24 feet = (W) 1 20 lbs. x 2 feet (240 foot-pounds) {240 foot-pounds). lere supposed to be applied in a direction parallel to the plane. a direction at an angle to the plane, as in Fig. 122, page 308, lolved into its components, parallel and perpendicular to the the line ^ <;' parallel to the incline; then the power applied, ;nt to the force actually expended & 1!, and to the pressure )n c c". The consumption of power is expressed by the pro- irallel equivalent, b c, into the length of tiie plane. Taking, LS above, the weight, 120 lbs., and the active power, 10 lbs., ly the parallel force b tf; then the amount of the horizontal )ower applied, b c, is found by proportion, thus AC:ABi:i/:if; arallel and horizontal forces are to each other as the base to the incline. WORK IN MOVING BODIES. 315 Work done with the Wedge. Supposing the wedge driven by a constant pressure through a distance equal to its length, the work done by the power is expressed by the power into the length, and the work done on the weight is expressed by the pro- duct of the weight into the breadth of the wedge. By equality of work, PxL = WxB, as before stated, in expressing equality of moments. If the wedge be driven for only a part of its length, the work done by the power is in the proportion of the part of the length driven; and the work done on the weight is similarly in the proportion of the part of the breadth by which the resisting surfaces are separated. Work done with the Screw. In one revolution of the screw, the weight is raised through a height equal to the pitch of the thread, whilst the power acts through the circum- ference of the circle described by the point at which it is applied to a lever. The products of the power and the weight by the spaces (^escribed by them are equal, or Px6.28r = Wx/, as before stated (page 311) to express equality of moments. Work done by Gravity. The work done by gravity on a falling body is equal to the weight of the body multiplied by the height through which it falls. Work accumulated in Moving Bodies. The quantity of work stored in a body in motion is the same as that which would be accumulated in it by gravity if it fell from such a height as would be sufficient to give it the same velocity; in short, from the height due to the velocity. (See Graviit, page 2^^),^ The accumulated work expressed in foot-pounds, is equal to the height so found in feet, multiplied by the weight of the body in pounds. The height due to the velocity is equal to the square of the velocity divided by 64.4, and the work and the velocity may be found directly from each other, according to the following rules: — Rule i. Given the weight and velocity of a moving body, to find the work accumulated in it. Multiply the weight in pounds by the square of the velocity in feet per second, and divide by 64.4. The quotient is the accumulated work in foot-pounds. Or, putting U for the work, v for the velocity, and 7v for the weight, U = 4^ (I) 64.4 Or, secondly: — Multiply the weight in pounds by the height in feet due to the velocity. The product is the accumulated work in foot-pounds. Or, putting A for the height, U = «/x// ( i^) 3l6 FUNDAMENTAL MECHANICAL PRINCIPLES. Work done by Percussive Force. If a wedge be driven by blows or strokes of a hammer or other heavy mass, the effect of the percussive force is measured by the quantity of work accumulated in the striking body. This work is calculated by the preceding rules, from the weight of the body and the velocity with which the blow is delivered, or directly from the height of the fall, if gravity be the motive power. The useful work done through the wedge is equal to the work delivered upon the wedge, supposing that there is no elastic or vibrating reaction from the blow, just as if the work had been delivered by a constant pres- sure equal to the weight of the striking body, exerted through a space equal to the height of the fall, or the height due to its final velocity. Of course, in order to give effect to the constant pressure on the wedge, now imagined to be brought into action, the pressure would require to be applied to the resisting medium through some combination of the mechanical elements. But where elastic action intervenes, a portion of the work delivered is uselessly absorbed in elastically straining the resisting body; and the elastic action may be, in some situations, so excessive as to absorb the whole of the work delivered. In this case, there would not be any useful work done. These remarks, applied to the action of a blow on a wedge, are applicable equally to the action of a blow of the monkey of a pile-driver upon a pile. If there be no elastic action, the work delivered being the product of the weight of the monkey by the height* of its fall, is equal to the work done in sinking the pile; that is, to the product of the frictional and other resistance to its descent by the depth through which it descends for one blow of the monkey. Supposing that the pile rests upon and is absolutely resisted by a hard unyielding obstacle, the work done becomes wholly useless, and consists of elastic or vibrating action ; or it may be that the head of the pile is split open. HEAT. THERMOMETERS. The action of Thermometers is based on the change of volume to which bodies are subject with a change of temperature, and they serve, as their name implies, to measure temperature. Thermometers are filled with air, water, or mercury. Mercurial thermometers are the most convenient, because the most compact. They consist of a stem or tube of glass, formed with a bulbous expansion at the foot to contain the mercury, which expands into the tube. The stem being uniform in bore, and the apparent expansion of mercury in the tube being equal for equal increments of temperature, it follows that if the scale be graduated with, equal intervals, these will indi- cate equal increments of temperature. A sufficient quantity of mercury having been introduced, it is boiled to expel air and moisture, and the tube is hermetically sealed. The freezing and the boiling points on the scale are then determined respectively by immersing the thermometer in melting ice and afterwards in the steam of water boiling under the mean atmospheric pressure, 14.7 lbs. per square inch, and marking the two heights of the column of mercury in the tube. The interval between these two points is divided into 180 degrees for Fahrenheit's scale, or 100 degrees for the Centigrade scale, and degrees of the same interval are continued above and below the standard points as far as may be necessary. It is to be noted that any inequalities in the bore of the glass must be allowed for by an adaptation of the lengths of the graduations. The rate of expansion of mercury is not strictly constant, but increases ¥dth the temperature, though, as already referred to, this irregularity is more or less nearly compensated by the varying rates of expansion of glass. In the Fahrenheit Thermometer, used in Britain and America, the number 0° on the scale corresponds to the greatest degree of cold that could be artificially produced when the thermometer was originally introduced. 32° ("the freezing-point") corresponds to the temperature of melting ice, and 212® to the temperature of pure boiling water — in both cases under the ordinary atmospheric pressure of 14.7 lbs. per square inch. Each division of the thermometer represents i® Fahrenheit, and between 32** and 212° there are i8o^ In the Centigrade Thermometer, used in France and in most other countries in Europe, o** corresponds to melting ice, and 100® to boiling water. From the freezing to the boiling point there are 100°. In the R^umur Thermometer, used in Russia, Sweden, Turkey, and Egypt, o** corresponds to melting ice, and 80° to boiling water. From the freezing to the boiling point there are 80°. 3 1 8 HEAT. Each degree Fahrenheit is | of a degree Centigrade, and y of a degree Reaumur, and the relations between the temperatures indicated by the different thermometers are as follows : — C. = I (F. -32). R. = I (F. -32). C. = f R. G. being the temperature in degrees Centigrade. R. do. do. Reaumur. F. do. do. Fahrenheit. That is to say, that Centigrade temperatures are converted into Fahrenheit temperatures by multiplying the former by 9 and dividing by 5, and adding 32° to the quotient; and conversely, Fahrenheit temperatures are converted into Centigrade by deducting 32°, and taking |ths of the remainder. Reaumur degrees are multiplied by | to convert them into the equivalent Centigrade degrees; conversely, |ths of the number of Centigrade degrees give their equivalent in Reaumur degrees. Fahrenheit is converted into Reaumur by deducting 32° and taking |ths of the remainder, and Reaumur into Fahrenheit by multiplying by f , and adding 32® to the product Tables No. 104, 105 contain equivalent temperatures in degrees Centigrade for given degrees Fahrenheit, from 0° F., or zero on the Falirenheit scale, to 608° F. ; and conversely, the temperature in degrees Fahrenheit correspond- ing to degrees Centigrade, from 0° C, or zero on the Centigrade scale, to 320° C. EQUIVALENT TEMPERATURES. 319 Table Na 104. — Equivalent Temperatures by the Fahrenheit AND Centigrade Thermometers. Degrees Degrees Degrees Degrees Degrees Degrees Degrees Fahr. Centigrade. Fahr. Centigrade. Fahr. Centigrade. Fahr. Centigrade. -17.78 + 38 + 3.34 + 76 + 24.45 + 114 + 45.56 + I 17.23 39 3.90 77 25.00 1^5 46.11 2 16.67 40 4.45 78 25.56 116 46.67 3 16.II 41 5.00 79 26.12 117 47.23 4 15.56 42 5.56 80 26.67 118 47.78 5 15.00 43 6. 1 1 81 I 27.23 119 48.34 6 14.45 44 6.67 82 27.78 120 48.90 7 13.90 45 7.23 ^3 28.34 121 49.45 8 13.34 46 7.78 84 28.89 122 50.00 9 12.78 47 8.34 85 29.45 123 50.56 10 12.23 48 8.89 86 30.00 124 5 I.I I II 11.67 49 9-45 87 30.55 125 51-67 12 II. II so 10.00 88 31. II 126 52.23 13 10.56 51 10.56 89 31.67 127 52.78 14 10.00 52 II. II 90 32.22 128 53-34 15 9.45 S3 11.67 91 32.78 129 53.90 16 ' 8.89 54 12.23 92 33-33 130 54.45 17 8.34 55 12.78 93 33.89 131 55-00 18 7.78 56 13.34 94 34.45 132 55-56 19 7.23 57 13.90 95 35.00 133 56.11 20 6.67 58 14.45 96 35.56 134 56.67 21 6. 1 1 59 15.00 97 36.11 135 57.23 22 5-56 60 15.56 98 36.67 136 57.78 23 5.00 61 16.11 99 37.23 137 58.34 24 4.45 63 16.67 100 37.78 138 58.90 25 3.90 63 17.23 lOI 38.34 139 59.45 26 3.34 64 17.78 102 38.90 140 60.00 27 2.78 65 18.34 103 39.45 141 60.56 28 2.23 66 18.89 104 40.00 142 61. II 29 1.67 67 19.45 105 40.56 143 61.67 30 I. II 68 20.00 106 41. II 144 62.23 31 0.56 69 20.56 j 107 41.67 145 62.78 32 0.00 70 21. II 108 42.23 146 63.34 33 + 0.56 71 21.67 109 42.78 147 63.90 34 I. II 72 22.23 no 43.34 148 64.45 35 1.67 73 22.78 , III 43.90 149 65.00 36 2.23 74 23-34 112 44.45 150 65.56 37 2.78 75 23.90 "3 45.00 151 66.11 HEAT. Table No. 104 {continued). ■^fX" Demes C«l«rade. ^^ Deerea ^S^." Cenugrade. f5^ Dcgr«s Cenugndt. 66.67 + 193 + 89.45 + 234 ■HI 12.23 + 275 + 135-00 67-^3 194 90.00 235 112.78 376 I35.56 67.78 195 90.56 236 "3-34 277 136" 68.34 196 91. II 237 113.90 278 136.67 6S.90 197 91.67 238 "4-45 279 137-23 69-45 198 92-23 239 115.00 380 137-78 70.00 "99 93.78 240 115.56 381 138.34 70.56 300 93-34 241 116.11 383 138.90 71.11 201 93-90 242 116.67 2S3 139.45 71.67 Z02 94-45 243 117-23 284 140.00 73.33 203 95-O0 344 117.78 285 140.56 73.78 304 95.56 245 118.34 386 141. II 73-34 205 96.11 246 118.90 287 141.67 73-9° 206 96.27 247 "9-45 288 142.23 74-45 207 97-23 248 120.00 389 142.78 75-00 208 97.78 249 120.56 390 143.34 75-56 209 98-34 250 121.11 291 133-90 76.1. 98.90 25" 131.67 292 144.45 76.67 211 99-4S 252 132.23 293 I45-00 77-^3 312 100.00 253 122.78 394 145-56 77.78 213 100.S6 254 '23.34 295 146. 1. 78.34 214 lOI.II 123-90 296 146.67 78.90 315 101.67 256 124-45 397 147.23 79-45 zi6 102.23 257 125.00 398 147.78 217 102.78 258 125.56 299 148.34 80.56 3l8 103-34 259 136.11 300 148.90 81. II 219 103.90 260 126.67 301 149-45 81.67 104-45 261 127-23 302 150.00 83.23 221 105.00 262 137.78 303 150-56 83.78 232 105.56 363 128.34 304 151.11 8334 223 106. 1 1 264 128.90 305 151.67 83.90 224 106.67 26s 129.45 306 152.23 84.4s 22s 107.23 266 130.00 307 152.78 85.00 336 107.78 267 130.56 308 153.34 85.56 227 108.83 268 131-11 309 '53.90 86.11 328 108.90 369 131-67 310 154-45 86.67 229 109.45 270 132.23 3" 155.00 8723 230 371 132-78 312 J5S-56 87.78 231 "a?6 272 133-34 313 88.34 23a iii.ii 273 133-90 314 Ise'eS 88.90 "33 111.67 274 134-45 3«S 157-23 EQUIVALENT TEMPERATURES. 321 Table No. 104 {continued). Fahrenheit and Centigrade. D^re«s De^ees Degrees Decrees Degrees De^ees Degrees Fahr. Degrees Fahr, Centigrade. Fahr. Centigrade. Fahr. Cenugrade. Centigrade. + 316 + 157.78 + 357 + 180.56 + 398 + 203.34 + 439 + 226.11 317 15^-34 358 181. II 399 203.90 440 226.67 318 158.90 359 181.67 400 204.45 441 227.23 319 159.45 360 182.23 401 205.00 442 227.78 320 160.00 361 182.78 402 205.56 443 228.34 321 160.56 362 183.34 403 206.11 444 228.90 322 161. II 363 183.90 404 206.67 445 229.45 323 161.67 364 184.45 405 207.23 446 230.00 324 162.23 365 185.00 406 207.78 447 230.56 325 162.78 366 185.56 407 208.34 448 231. II 326 163.34 367 186. 1 1 408 208.90 449 231.67 327 163.90 368 186.67 409 209.45 45^ 232.23 328 164.45 369 187.23 410 210.00 451 232.78 329 165.00 370 187.78 411 210.56 452 233.34 330 165.56 371 188.34 412 211. II 453 233.90 331 166.II 372 188.90 413 211.67 454 234.45 332 166.67 373 189.45 414 212.23 455 235.00 333 167.23 '374 190.00 415 212.78 456 235.56 334 167.78 375 190.56 416 213.34 457 236.11 335 168.34 376 191. II 417 213.90 458 236.67 336 168.90 377 191.67 418 214.45 459 237.23 337 169.45 378 192.23 419 215.00 460 237.78 338 170.00 379 192.78 420 215.56 461 238.34 339 170.56 380 193.34 421 2l6.II 462 238.90 340 171. II 381 193.90 422 216.67 463 239.45 341 171.67 382 194.45 423 217.23 464 240.00 342 172.23 383 195.00 424 217.78 465 240.56 343 172.78* 384 195.56 425 218.34 466 241. II 344 173.34 385 I96.II 426 218.90 467 241.67 345 173.90 386 196.67 427 219.45 468 242.23 346 174.45 387 197.23 428 220.00 469 242.78 347 175.00 388 197.78 429 220.56 470 243.34 348 • 175.56 389 198.34 430 221. II 471 243.90 349 1 76. 1 1 390 198.90 431 221.67 472 244.45 350 176.67 391 199.45 432 222.23 473 245.00 351 177.23 392 200.00 433 222.78 474 245.56 352 177.78 393 200.56 434 223.34 475 246.11 353 178.34 394 201. II 435 223.90 476 246.67 354 178.90 395 201.67 436 224.45 477 247.23 355 179.45 396 202.23 437 225.00 478 247.78 356 > 180.00 397 202.78 438 225.56 479 248.34 21 Table No. 104 (cotUitnud). 'ahrenmeit and Ckntigrao Ccnli^ijdc DegMS Centigrade, ■iX" Degrees Cemigrade. '^^ C™.«™ic, + 248.90 + 5"3 + 267.23 + 546 + 285.56 + 579 + 303-90 249-45 SI4 267-78 547 286.11 5B0 304-45 250.00 515 268-34 548 286.67 58' 305.00 250.56 516 268-90 549 287.23 582 305-5'' 25I-" 5"7 269.45 55° 287.78 583 306.11 2S,.67 5>8 270.00 55' 288.34 584 306.67 252.23 5>9 270.56 552 288.90 585 307-23 252.78 520 271.11 553 289.45 586 307.78 253.34 521 271.67 554 290.00 S87 308.34 253-90 522 272.23 555 290.56 588 308.90 254-45 523 272.78 556 291. II 589 309-45 255.00 524 273-34 557 291.67 590 310.00 255.56 525 27390 558 292.23 591 310.56 256.,. 526 27445 559 292.78 592 311. II 256.67 527 275.00 560 29334 593 311.67 257-23 528 275-5' 561 293.90 594 312.23 257.78 529 276-1. 562 294.45 595 312.78 258-34 530 276.67 563 295.00 596 313-34 25890 531 277-23 564 295-56 597 313-90 25945 532 277-78 565 296.11 598 314-45 260-00 533 278.34 566 296.67 599 315-00 260-56 534 278.90 567 297.23 600 315-56 261-11 535 279.45 568 297.78 601 316.11 261.67 536 280.00 569 298.34 602 316.67 262.23 537 280.56 570 298.90 603 317-23 262.78 538 281.11 571 299-45 604 31778 263.34 539 281.67 572 300.00 605 3T8.34 263.90 540 282.23 573 300.56 606 318.90 264.45 541 282.78 574 301. 11 607 319-45 265.00 542 283-34 575 301.67 608 320.00 265-56 543 283-90 576 302.23 266.11 544 284-45 577 302.78 366.67 545 285.00 578 303.34 EQUIVALENT TEMPERATURES. 323 Table No. 105. — Equivalent Temperatures by the Centigrade and Fahrenheit Thermometers. Degrees Degrees Degrees Degrees Fahr. Degrees Degrees Degrees Degrees Cent. Fahr. CenL Cent. Fahr. Cent. Fahr. -20 - 4.0 + 21 + 69.8 + 62 + 143-6 + 103 + 217.4 19 2.2 22 71.6 63 145.4 104 219.2 18 0.4 23 73-4 64 147.2 105 22l!o 17 + 1.4 24 75.2 65 149.0 106 222.8 16 3.2 ^5 77.0 66 150.8 107 224.6 15 5.0 26 78.8 67 152.6 108 226.4 H 6.8 27 80.6 68 154.4 109 228.2 13 8.6 28 82.4 69 156.2 no 230.0 12 10.4 29 84.2 70 158.0 III 231.8 II 12.2 30 86.0 71 159.8 112 233.6 10 14.0 31 87.8 72 161.6 "3 235-4 9 15.8 32 89.6 73 163.4 114 237.2 8 17.6 33 91.4 74 165.2 "5 239.0 7 19.4 34 93-2 75 167.0 116 240.8 6 21.2 35 95.0 76 168.8 117 242.6 5 23.0 36 96.8 77 170.6 118 2444 4 24.8 ^l 98.6 78 172.4 119 246.2 3 26.6 38 100.4 79 174.2 120 248.0 2 28.4 39 102.2 80 176.0 121 249.8 I 30.2 40 104.0 81 177.8 122 251.6 32.0 41 105.8 82 179.6 123 253.4 + I 33.8 42 107.6 83 181.4 124 255.2 2 35.6 43 109.4 84 183.2 125 257.0 3 37.4 44 III. 2 85 185.0 126 258.8 4 39-2 45 1 130 86 186.8 127 260.6 5 41.0 46 1 14.8 87 188.6 128 262.4 6 42.8 47 1 16.6 88 190.4 129 264.2 7 44.6 48 1 18.4 89 192.2 130 266.0 8 46.4 49 120.2 90 194.0 131 267.8 9 48.2 50 122.0 91. 195.8 132 269.6 10 50.0 51 123.8 92 197.6 133 271.4 II 51.8 52 125.6 93 199.4 134 273.2 12 53.6 53 127.4 94 201.2 135 275.0 13 55-4 54 129.2 95 203.0 136 276.8 14 57.2 55 1310 96 204.8 137 278.6 15 59.0 56 132.8 97 206.6 138 280.4 16 60.8 57 134.6 98 * 208.4 139 282.2 17 62.6 58 136.4 99 210.2 140 284.0 18 64.4 59 138.2 100 212.0 141 285.8 19 66.2 60 140.0 lOI 213.8 142 287.6 20 68.0 61 141.8 102 215.6 143 289.4 324 HEAT. Table No. 105 {continued). Centigrade and Fahrenheit. Depves Jjcgtccs Degrees Degrees Degrees. DcgFMS Cent. Fahr. CenL Fahr. C«lL Fahr. Cent. Fahr. + 144 + 291.2 + 189 + 372.2 1 + 234 + 453.2 + 279 + 534-2 H5 293.0 190 . 374.0 235 455.0 280 536.0 146 294.8 191 375-8 236 456.8 281 537.8 147 296.6 192 377.6 237 458.6 282 539.6 148 298.4 193 379.4 238 460.4 283 541.4 149 300.2 194 381.2 239 462.2 284 543.2 150 302.0 195 383.0 240 464.0 285 545.0 151 303-8 196 384.8 . 241 465.8 286 546.8 152 305.6 '97 386.6 242 467.6 287 548.6 153 307.4 198 388.4 243 469.4 288 1 5504 154 309.2 199 390.2 244 471.2 i 289 552.2 '55 31 10 200 392.0 245 473.0 I 290 554.0 156 312.8 201 393.8 246 474.8 291 555.8 157 314.6 202 395-6 247 476.6 292 557.6 158 316.4 203 397.4 248 478.4 293 559.4 159 318.2 204 399-2 249 480.2 : 294 561.2 160 320.0 205 401.0 250 482.0 i 295 563.0 161 321.8 206 402.8 251 483.8 > 296 564.8 162 3236 207 404.6 252 485.6 297 566.6 163 325.4 208 406.4 253 487.4 298 568.4 164 327.2 209 408.2 254 489.2 299 570.2 '^l 329.0 210 410.0 255 491.0 300 572.0 166 330.8 211 41 1.8 256 492.8 301 573.8 167 332.6 212 413.6 257 494.6 302 575.6 168 334-4 213 415-4 258 496.4 303 577.4 169 336.2 ' 214 417.2 259 498.2 304 579.2 170 338.0 215 419.0 260 500.0 305 581.0 171 339-8 216 420.8 261 501.8 306 582.8 172 341.6 217 422.6 262 503.6 307 584.6 173 343.4 218 424.4 263 505.4 308 586.4 174 345.2 219 426.2 264 507.2 309 588.2 175 347.0 220 428.0 265 509.0 310 590.0 176 348.8 221 429.8 266 510.8 311 591.8 177 350.6 222 431-6 267 512.6 312 593.6 178 352.4 223 433.4 268 514.4 313 595-4 179 354.2 ' 224 435.2 269 516.2 314 597.2 180 356.0 225 437.0 270 518.0 315 599.0 181 357.8 226 438.8 271 519.8 316 600.8 182 359.6 227 440.6 272 521.6 317 602.6 183 361.4 228 442.4 273 523.4 318 604.4 184 363.2 229 444.2 274 525.2 319 606.2 185 365.0 230 446.0 275 527.0 320 608.0 186 366.8 231 447.8 276 528.8 187 368.6 232 449.6 277 530.6 188 370.4 233 451.4 278 532.4 AIR-THERMOMETERS. 32s C t Air-Thermometers. Air-thermometers, or gas-thermometers, though inconvenient because bulky, are, by reason of the great expansiveness of air, superior to such as depend upon the expansion of liquids or solids, in point of delicacy and exactness. In any thermometer, whether liquid or gas, the indications depend jointiy upon the expansion by heat of the fluid substance, and that of the tube which holds it The expansion of mercury is scarcely seven times that of the glass tube within which it expands, and the exactness of its indications are interfered with by the variation in the expansiveness of glass of different qualities. In the gas-thermometer, on the contrary, the expansiveness of the gas is 160 times that of the glass, and the inequalities of the glass do not sensibly affect the indications of the instrument Gas-thermometers, or, as they are commonly called, air-thermometers, are designed either to maintain a constant pressure with a varying volume of air, or to maintain a constant volume of air while the pressure varies. In the first case. Fig. 119, the thermometer consists of a reservoir a, to be placed in the substance of which the temperature is to be ascertained; a tube d/, connected at a suitable distance by a small tube alf to the reservoir; a tube cd, open above, through which mercury is introduced into the instrument; a stop-cock r to open or close a communication — ist, between the tube dfzjoA the atmosphere; 2d, between the base of the tube cd and the atmosphere; 3d, between the two tubes df^ cd\ 4th, between both these tubes and the atmosphere. The tube df^ which is carefully gauged, answers the purpose of the gradu- ated tube of the mercury-thermometer, and receives the air driven over by expansion from the reservoir, at the same time that it is maintained at or near the temperature of the surrounding atmosphere. Thus the air is divided between the reservoir a and the tube df, of which the air in the former is at the ^* "9.-Air-Thcnnomcter. temperature of the substance under observation, and that in the latter is at the temperature of the atmosphere. These two portions of air support the same pressure, which qm at all times be approximated to that of the atmosphere by means of the cock r, through which the mercury is allowed to escape until it arrives at the same level in the two tubes. By means of a formula embracing the respective volumes of the two portions of air and the temperature of the atmosphere, the temperature of the substance under observation is determined. But it is apparent that, when applied as a pyrometer to the measurement of high temperatures — ^higher, that is to say, than the boiling point of mercury (676° F.) — ^the greater part of the air passes by expansion into the tube df^ leaving but a small remainder in the reservoir a. A serious objection to this is that the proportion of air which passes over into the tube df for a new increase of temperature is very small, and is with difficulty measured with sufficient precision. The second form of air thermometer, in which the pressure varies whilst the volume remains the same, was used by M. Regnault in his researches. 326 HEAT. f The temperature is measured by means of the increased elastic force of the inclosed air, and the instrument is both more convenient and more precise than that in which the volume varies, for at all temperatures the sensibility of the instrument is the same. At high temperatures the apparatus is liable to distortion under the pressure of the inclosed air; but this may be pre- vented, if needful, by introducing air of a lower than atmospheric- pressure at an ordinary temperature, even so low as one-fourth of an atmosphere; for, although the apparatus is less sensitive in proportion as the first supply of air is of less density and pressure, yet withal it is sufficiently sensitive. The thermometer, as employed by M. Regnault, is shown in Fig. 120. Two glass tubes, df^ cd, about half-an-inch bore, are united at the base by a stop-cock r. The tube cd is open above, and df is con- nected to the reservoir a by a small tube ab. The cover of the boiler in which the reser- voir is inclosed is shown at b, and the tubes are protected from the heat of the boiler by the partition c d. By means of a three-way connection, g^ and tube hy the connecting tube ab communicates with an air pump, by means of which the apparatus may be dried, and air or other gas supplied to it The first thing to be done is to completely dry the apparatus, and for this object, a little mercury is passed into the tube bd^ and the cock r is closed against it. The exhausting pump is then set to work to exhaust the tube, which is done several times, the air being slowly re-admitted after each exhaus- tion, after having been passed through a filter of pumice-stone in connection with the pump, saturated with concentrated sul- phuric acid to absorb moisture, and thus desiccate the air. During this part of the process, the reservoir is maintained at a temperature of 130° F., or 140° F., to insure complete desiccation. Next, the reservoir is plunged into melting ice, the two vertical tubes bd^ cd, are put into conynunication, and filled with mercury up to a suitable level /, marked on the tube bd. If it is desired to establish an internal pressure less than that of the atmosphere, the air is partially exhausted by means of the pump, the degree of exhaustion being recorded by the difference of level in the two tubes. The exhausting tube h is then hermetically sealed, and the mercury adjusted to the level /in the tube bd. G Fig. Z20. Pyrometers. Pyrometers are employed to measiu-e temperatures above the boiling point of mercury, about 676* F. They depend upon the change of form of either solid or gaseous bodies, liquids being necessarily inadmissible. Pyrometric estimations are of three classes : — First, those of which the PYROMETERS. 32/ indications are based upon the change of dimensions of a particular body, solid or gaseous — the pyrometer; second, those based on the heat imparted to water by a heated body; third, those which are based upon the melting points of metals and metallic alloys. Wedgwood^s pyrometer, invented in 1782, was founded on the property possessed by clay of contracting at high temperatures, an eflfect which is due partly to the dissipation of the water in clay, and subsequently to partial vitrification. The apparatus consists of a metallic groove, 24 inches long, the sides of which converge, being half-an-inch wide above and three-tenths below. The clay is made up into little cylinders or truncated cones, which fit the commencement of the groove after having been heated to low red- ness; their subsequent contraction by heat is determined by allowing them to slide from the top of the groove downwards till they arrive at a part of it through which they cannot pass. The zero point is fixed at the tempera- ture of low redness, 1077° F. The whole length of the. groove or scale is divided into 240 degrees, each of which was supposed by Wedgwood equivalent to 130° F., the other end of the scale being assumed to represent 32,277° F. Wedgwood also assumed that the contraction of the clay was proportional to the degree of heat to which it might be exposed; but this assumption is not correct, for a long-continued moderate heat is found to cause the same amount of contraction as a more violent heat for a shorter period. Wedgwood's pyrometer is not employed by scientific men, because its indications cannot be relied upon for the reason just given, and also because the contraction of different clays under great. heat is not always the same. In Daniell's pyrometer the temperature is measured by the expansion of a. metal bar inclosed in a black-lead earthenware case, which is drilled out longitudinally to ^ inch in diameter and 7^ inches deep. A bar of platinum or soft iron, a little less in diameter, and an inch shorter than the bore, is placed in it and surmounted by a porcelain index i J^ inches long, kept in its place by a strap of platinum and an earthenware wedge. When the instrument is heated, the bar, by its greater rate of expansion compared with the black-lead, presses forward the index, which is kept in its new situation by the strap and wedge until the instrument cools, when the observation can be taken by means of a scale. The air-pyrometer. The principle and construction of the air-thermo- meter are directly applicable for pyrometric purposes, substituting a platinum globe for the glass reservoir already described, for resisting great heat, and as large as possible. The' chief cause of uncertainty is the expansion of the metal at high temperatures. The second means of estimation is best represented by the " pyrometer " of Mr. Wilson, of St. Helen's. He heats a given weight of platinum in the fire of which the temperature is to be measured, and plunges it into a vessel containing twice the weight of water of a known temperature. Observing the rise of temperature in the water, he calculates the tempera- ture to which the platinum was subjected, in terms of the rise of tempera- ture of the water, the relative weights of the platinum and the water, and their specific heats. In fact, the elevation of the temperature of the water is to that of the platinum above the original temperature of the water in the compound ratio of the weights and specific heats inversely; that is to say, that the weights of the platinum and the water being as i to 2, and 328 HEAT. their specific heats as .0314 to i, the rise of temperature of the water is to that of the platinum as i x .0314 to 2 x i, or as i to 63.7, and the rule for finding the temperature of the fire is to multiply the rise of temperature of the water by 63. 7, and add its original temperature to the product. The sum is the temperature of the fire, subject to correction for the heat absorbed by the thermometer in the water, and by the iron vessel contain- ing the water, and the heat retained by the platinum. The correction is estimated by Mr. Wilson at iV^j taking the weight of water at 2000 grains, and that of the platinum 1000 grains, and it may be allowed for by increas- ing the above-named multiplier by -rrth, to 67.45. Mr. Wilson proposed that for general practical purposes a small piece of Stourbridge clay be substituted for platinum, to lessen the cost of the apparatus. With a piece of such clay, weighing 200 grains, and 2000 grains of water, he found that the correct multiplier was 46. The third means of estimation, based on the melting points of metals and metallic alloys, is applied simply by suspending in the heated medium a piece of metal or alloy of which the melting point is known, and, if necessary, two or more pieces of different melting points, so as to ascertain, according to the pieces which are melted and those which continue in the solid state, within certain limits of temperature, the heat of the furnace. A list of melting points of metals and metallic alloys is given in a subsequent chapter. Luminosity at High Temperatures. The luminosity or shades of temperature have been observed by M. Pouillet by means of an air-pyrometer to be as follows : — Shaob. Tempsraturs, Temperature, Centigrade. Fahrenheit. Nascent Red 525° 977° Dark Red 700 1292 Nascent Cherry Red 800 1472 Cherry Red 900 1652 Bright Cherry Red 1000 1832 Very Deep Orange iioo 2012 Bright Orange. 1200 2192 White 1300 2372 "Sweating" White 1400 2552 Dazzling White 1500 2732 A bright bar of iron, slowly heated in contact with air, assumes the following tints at annexed temperatures (Claudel) : — Centigrade. Fahrenheit 1. Cold iron at about 12*' or 54° 2. Yellow at 225 437 3. Orangeat 243 473 4. Red at 265 509 5. Violetat 277 531 6. Indigo at 288 550 7- Blue at 293 559 8. Green at 332 630 9. Oxide Gray (gris cToxyde) at 400 752 MOVEMENTS OF HEAT. 329 MOVEMENTS OF HEAT. When two bodies in the neighbourhood of each other have unequal temperatures, there exists between them a transfer of heat from the hotter of the two to the other. The tendency to an equalization, or towards an equilibrium, of temperatures in this way is universal, and the passage of heat takes place in three ways : by radiation, by conduction, and by con- vection or carriage from one place to another by heated currents. Radiation of Heat from Combustibles. It is a common assumption that the radiation of heat from combustibles is relatively very small in comparison with the total quantity of heat evolved. Holding the hand near the flame of a candle, laterally, the radiant heat, which is the only heat thus experienced, is much less than the heat experienced by the hand when held above the flame, which is the heat by convection of the hot current of air which rises from the flame. But it is to be noted that, whilst the radiant heat is dissipated all round the flame, the diameter of the upward current is little more than that of the flame, and the conveyed heat is therefore concentrated in a small compass. M. Peclet, by means of a simple apparatus, consisting of a cage suspend- ing the combustible within a hollow cylinder filled with water in an annular space, ascertained that the proportion of the total 'heat radiated from clifrerent combustibles was as follows : — Radiant heat from wood nearly }(. Do. do. wood charcoal „ J?. Do. do. oil „ '/j. These values serve to show that radiation of heat is considerable, and that flameless carbon radiates much more than flame, though the proportion of heat radiated from fuels depends very much upon the disposition of the material and the extent of radiating surface. With respect to heated bodies, apart from combustibles as such, the radiation or emission of heat implies the reverse process of absorption, and the best radiators are likewise the best absorbents of heat. All bodies possess the property of radiating heat The heat rkys proceed in straight lines, and the intensity of the heat radiated from any one source of heat becomes less as the distance from the source of heat increases, in the inverse ratio of the square of the distance. That is to say, for example, that at any given distance from the source of radiation, the intensity of the radiant heat is four times as great as it is at twice the distance, and nine times as great as it is at three times the distance. The quantity of heat emitted by radiatioh increases in some proportion with the difference of temperatures of the radiating body and the surrounding medium, but more rapidly than the simple proportion for the greater differ- ences; and the quantity of heat, greater or less, emitted by bodies by radiation under the same circumstances is the measure of their radiating p<nver. Radiant heat traverses air without heating it. When a polished body is struck by a ray of heat, it absorbs a part of the heat and reflects the rest The greater or less proportion of heat absorbed by the body is the measure of its absorbing power, and the reflected heat is the measure of its reeding power. 330 HEAT. When the temperature of a body remains constant it indicates that the quantity of heat emitted is equal to the quantity of heat absorbed by the body. The reflecting power of a body is the complement of its absorbing power; that is to say, that the sum of the absorbing and reflecting powers of all bodies is the same, which amounts to this, that a ray of heat striking a body is disposed of by absorption and reflection together, that which is not absorbed being necessarily reflected. For example, the radiating power of a body being represented by 90, the reflecting power is also 90, and the absorbing power is lo, supposing that Table No. 106. — Comparative Radiating or Absorbent and Reflecting Powers of Substances. Substance. Lamp Black Water Carbonate of Lead Writing Paper Ivory, Jet, Marble Isinglass Ordinary Glass China Ink Ice Gum Lac Silver Leaf on Glass Cast Iron, brightly polished Mercury, about Wrought Iron, polished Zinc, polished Steel, polished Platinum, a little polished Do. deposited on Copper . . . Do. in Sheet Tin Brass, cast, dead polished Do. hammered, dead polished., Do. cast, bright polishefl Do. hammered, bright polished Copper, varnished Do. deposited on iron Do. hammered or cast Gold, plated Do. deposited on polished Steel Silver, hammered, polished bright Do. cast, polished bright Powers. Radiating or Absorbing. Reflecting. 100 100 100 98 2 93 to 98 7 to 2 91 9 90 10 85 15 85 15 72 28 27 73 25 75 23 77 23 77 19 Si 17 ' 83 24 76 17 83 17 83 15 85 II 89 9 91 7 93 7 93 14 86 7 93 7 93 5 95 3 97 3 97 3 97 MOVEMENTS OF HEAT. 331 the total •quantity of heat which strikes the body is represented by 100. The reflecting power of soot is sensibly «/7, and its absorbing and radiating powers are 100. The absorbing power varies with the nature of the source of heat, with the condition of the substance, and with the inclination of the direction of the heat radiated upon the body. That of a metallic surface is so much the less, and consequently the reflecting power is so much the more, in proportion as the surface is better polished. The reflecting power of metals, according to MM. de la Provostaye and Desains, is practically the same, when the angle of incidence, that is the angle at which the rays of heat strike the surface, is less than 70® of inclina- tion with the surface; but for greater angles, approaching more nearly to 90°, perpendicular to the surface, it sensibly diminishes. For example, at angles of from 75 to 80 degrees, the reflecting power is only 94 per cent, of what it is under the smaller angles of incidence. The table No. 106 contains the radiating and absorbing powers and the reflecting powers of various substances. (Leslie^ De la Provostaye and JDesainSy and Melloni,) The reflecting power of glass has been found to be the same for heat and for light Conduction of Heat. — Conduction is the movement of heat through sub- stances, or from one substance to another in contact with it. The table No. 107 contains the relative internal conducting power of metals and earths, according to M. Despretz. A body which conducts heat well is called a good conductor of heat; if it conducts heat slowly, it is a bad conductor of heat. Bodies which are finely fibrous, as cotton, wool, eider-down, wadding, finely divided charcoal, are the worst conductors of heat. Liquids and gases are bad conductors; but if suitable provision be made for the free circulation of fluids they may abstract heat very quickly by contact with heated surfaces, acting by convection. Convection of Heat, — Convected or carried heat is that which is trans- ferred from one place to another by a current of liquid or gas : for example, by the products of combustion in a furnace towards the heating surface in the flues of a boiler. Table No. 107. — Relative Internal Conducting Power of Bodies. Substance. Relative conducting power. t Substance. Relative conducting power. Gold 1000 981 973 892 749 562 374 Zinc 363 304 180 24 12 II Platinum ; Tin Silver Lead Copper Marble Brass Porcelain Cast Iron Terra Cotta Wrought Iron THE MECHANICAL THEORY OF HEAT. ' Heat and mechanical force are identical and convertible. Independently of the medium through which heat may be developed into mechanical action, the same quantity of heat is resolved into the same total quantity of work. The English unit of heat is that which is required to raise the temperature of i lb. of water r degree Fahr. If 2 lbs. of water be raised 1 degree, or i lb. be raised 2 degrees in temperature, the expenditure of f heaCj and to express the lies between the unit of le foot-pound, on the other ; of the numerical relation s obtained by the following iptator. Fig. izi, consisting of a vertical shaft carry- ing a brass paddle-wheel, of which the paddles re- volved between station- ary vanes, which served to prevent the liquid in the vessel from being bodily whirled in the direction of rotation. The vessel was filled with water, and the agi- tator was made to revolve bymeans of a cord wound round the upper part of the shaft, and attached to a weight which de- scended m front of a scale, by which the work done was measured. found that the heat com- ited to one pound-degree expended in producing it. at was capable of raising jnical equivalent of heat, ;n as 772 foot-pounds for nedium, and it yielded die alent for different thermo- bot-pounds, ;s(say424)tilogrammetres. .60 (say 1390) foot-pounds, [lish thermal units — about MECHANICAL THEORY OF HEAT. 333 According to the mechanical theory of heat, in its general form, heat, mechanical force, electricity, chemical affinity, light, and sound, are but different manifestations of motion. Dulong and Gay Lussac proved by their experiments on sound, that the greater the specific heat of a gas, the more rapid are its atomic vibrations. Elevation of temperature does not alter the rapidity but increases the length of their vibrations, and in con- sequence produces "expansion " of the body. All gases and vapours are assumed to consist of numerous small atoms, moving or vibrating in all directions with great rapidity; but the average velocity of these vibrations can be estimated when the pressure and weight of any given volume of the gas is known, pressure being, as explained by Joule, the impact of those numerous small atoms striking in all directions, and against the sides of the vessel containing the gas. The greater the number of these atoms, or the greater their aggregate weight, in a given space, and the higher the velocity, the greater is the pressure. A double weight of a perfect gas, when con- fined in the same space, and vibrating with the same velocity — that is, having the same temperature — gives a double pressure; but the same weight of gas, confined in the same space, will, when the atoms vibrate with a double velocity, give a quadruple pressure. An increase or decrease of temperature is simply an increase or decrease of molecular motion. When the piston in the cylinder yields to the pressure of steam, the atoms will not rebound from it with the same velocity with which they strike, but will return after each succeeding blow, with a velocity continually decreasing as the piston continues to recede, and the length of the vibrations will be diminished. The motion gained by the piston will be precisely equivalent to the energy, heat, or molecular motion lost by the atoms of the gas; and it would be as reasonable to expect one billiard ball to strike and give motion to another without losing any of its own motion, as to suppose that the piston of a steam-engine can be set in motion without a corresponding quantity of energy being lost by some other body. In expanding air spontaneously to a double volume, delivering it, say, into a vacuous space, it has been proved repeatedly that the air does not appreciably fall in temperature, no external work being performed; but that, on the contrary^ if the air at a temperature, say, of 230° F., be expanded against an opposing pressure or resistance, as against the piston of a cylinder, giving motion to it and raising a weight or otherwise doing work, the tem- perature will fall nearly 170® F. when the volume is doubled, that is from 230° F. to about 60® F., and, taking the initial pressure at 40 lbs., the final pressure would be 15 lbs. per square inch. When a pound weight of air, in expanding, at any temperature or pressure, raises 130 lbs. one foot high,' it loses i® F. in temperature; in other words, this pound of air would lose as much molecular energy as would equal the energy acquired by a weight of one pound falling through a height of 130 feet. It must, however, be remarked that but a small portion of this work — 130 foot-pounds — can be had as available work, as the heat which disappears does not depend on the amount of work or duty realized, but upon the total of the opposing forces, including all resistance from any external source whatever. When air is compressed the atmosphere descends and follows the piston, assisting in the operation with its whole weight; and when air is expanded the motion of the piston is, on the contrary, opposed by the whole weight of the atmosphere, which is again raised. Although, 334 HEAT. therefore, in expanding air, the heat which disappears is in proportion to the total opposing force, it is much in excess of what can be rendered available; and, commonly, where air is compressed the heat generated is much greater than that which is due to the work which is required to be expended in compressing it, the atmosphere assisting in the operation. Let a pound of water, at a temperature of 212** F., be injected into a vacuous space or vessel, having 26.36 cubic feet of capacity — the volume of one pound of saturated steam at that temperature — ^and let it be evapor- ated into such steam, then 893.8 units of heat would be expended in the process. But if a second pound of water, at 212°, be injected and evapor- ated at the same temperature, under a uniform pressure of 14.7 lbs. per square inch, being the pressure due to the temperature, the second pound must dislodge the first, supposing the vessel to be expansible, by repelling 'that pressure; and this involves an amount of labour equal to 55,800 foot- pounds (that is, 14.7 lbs. x 144 square inches x 26.36 cubic feet), and an additional expenditure of 72.3 units of heat (that is, 55,800 -f- 772), making a total, for the second pound, of 965. i units. Similarly, when 1408 units of heat are expended in raising the tempera- ture of air under a constant pressure, 1000 of these units increase the velocity of the molecules, or produce a sensible increase of temperature ; while the remaining 408 units, which disappear as the air expands, are directly consumed in repelling the external pressure for the expansion of volume. Again, if steam be permitted to flow from a boiler into a comparatively vacuous space without giving motion to another body, the temperature of the steam entering this space would rise higher than that of the steam in the boiler. Or, suppose two vessels, side by side, one of them vacuous and the other filled with air at, say, two atmospheres; if a communication be opened between them, the pressure becomes the same in both. But the temperature would fall in one vessel and rise in the other; and although the air is expanded in this manner to double its first volume, there would not, on the whole, be any appreciable loss of heat, for if the separate por- tions of air be mixed together, the resulting average temperature of the whole would be very nearly the same as at first It has been proved experimentally, corroborative of this statement, that the quantity of heat required to raise the temperature of a given weight of air, to a given extent, is the same, irrespective of the density or the volume of the air. Regnault and Joule found that to raise the temperature of a pound of air, whether I cubic foot or 10 cubic feet in volume, the same quantity of heat was expended. In rising against the force of gravity steam becomes colder, and it par- tially condenses while ascending, in the effort of overcoming the resistance of gravity. For instance, a column of steam weighing, on a square inch of base, 250.3 lbs., that is to say, having a pressure of 250.3 lbs. per square inch, would, at a height of 275,000 feet, be reduced to a pressure of i lb. per square inch, and, in ascending to this height, the temperature would fall from 401° to 102° F., while, at the same time, nearly 25 per cent of the whole vapour would be precipitated in the form of water, unless it were supplied with additional heat while ascending. If a body of compressed air be allowed to rush freely into the atmosphere, the temperature falls in the rapid part of the current, by the conversion of EXPANSION BY HEAT. 335 heat into motion, but the heat is almost all reproduced when the motion has quite subsided. From recent experiments, it appears that nearly similar results are obtained from the emission of steam under pressure. When water falls through a gaseous atmosphere, its motion is constantly retarded as it is brought into collision with the particles of that atmosphere, and by this collision it is partly heated and partly converted into vapour. If a body of water descends freely through a height of 772 feet, it acquires from gravity a velocity of 223 feet per second; and, if suddenly brought to rest when moving with this velocity, it would be violently agitated, and would be raised one degree of temperature. But suppose a water-wheel, 772 feet in diameter, into the buckets of which the water is quietly dropped; when the water descends to the foot of the fall, and is delivered gently into the tail-race, it is not sensibly heated. The greatest amount of work it is possible to obtain from water falling from a given level to a lower level is expressible by the weight of water multiplied by the height of the fall. These illustrative exhibitions of the nature and reciprocal action of heat and motive power, show that the nature and extent of the change of tem- perature of a gas while expanding depend nearly altogether upon the cir- cumstances under which the change of volume takes place. EXPANSION BY HEAT. All bodies are expanded by the application of heat, but in different degrees. Expansion is measurable in three directions : — Length, breadth, and thickness; and it may be measured as linear expansion, in one direc- tion; as superficial expansion, in two directions; or as cubical expansion, in three directions. Linear expansion, or the expansion of length, is that which will be exposed in the following tables for solids and liquids. The expansion of gases is measured cubically, by volume. Superficial expansion, it may be added, is twice the linear expansion, and cubical expansion is three times the linear expansion. That is to say, the additional volume by expansion in two direc- , tions, as in length and breadth, is twice the I additional volume in one direction; and the : additional volume in three directions is three I times that in one direction. For example, j take a solid cube abcdefg\ the expansion in ' one direction ea^ on the face abed, is, say, equal to that indicated by the dot lines pro- jected from that face, and the volume by expansion is equal to the extension of the surface abed thus projected. In each of the two other directions, da, upwards, and ab, *• "^ laterally, the volume by. expansion is the same as that of the expansion on the face abed. Consequently, the total increase of volume by expansion, as measured cubically, in the three directions of length, breadth, and thickness, is three times the increase of volume in one direction singly; and, as measured superficially, in two of these directions, it is twice the increase of volume in one direction. 336 HEAT. Table No. io8. — Linear Expansion of Solids by Heat, between 32° AND 212*" F. METALS. } Zinc, sheet Do., forged Lead Zinc 8+1 tin, slightly ham mered, White Solder: — tin i -♦- 2 lead. Tin, grain Tin Silver Speculum metal Brass Copper Gun Metal: — 16 copper + i tin 8 copper + i tin Yellow Brass : — Rod Do. Trough form.. Gold:— Paris standard,' annealed Do. unannealed Bismuth Iron, forged Do. wire Steel, rod, 5 feet long Do. tempered Do. not tempered Cast Iron, rod, 5 feet long Antimony Palladium Platinum Expansion between m' and 213* F. in common fractions. 1/ ■/3s. ■/: ■/ ■/ 37a 399 V 403 524 753a V58X V5a4 Vsaa Vsa8 V. /66x 64s V719 V8x9 Vsia V874 y8o7 '/901 ^/9a3 /xooo 7x167 Expansion between 32* and 2x2' F. in a length =ioa length = xoo. 29416 31083 28484 26917 25053 24833 21730 19075 19333 18782 17220 19083 18167 18930 18945 I5153 I5516 I3917 12204 12350 1 1450 12396 10792 moo 10833 1 0000 08570 Expansion between 32' and 2X2* F. in a length' of xo feet. inch. •353 .374 .342 .322 .301 .298 .260 .229 .232 .225 .207 .229 .218 .227 .227 .181 .186 .167 .146 .148 •137 .149 .130 •133 .130 .120 .103 Expansion for I F. in a length of TOO fecL inch. .0196 .0207 .0190 .0179 .0167 .0166 .0145 .0127 .0130 .0125 .0115 .0127 .0121 .0126 .0126 .0101 .0103 .00928 .00814 .00823 .00763 .00826 .00719 .00740 .00722 .00667 .00571 From 0° to 300° C. (32° F. to 572° F.) Copper... Iron Platinum. / 0° to 100° C. ( 0° to 300° C. / 0° to loo** C. \ 0° to 300° C. { 0° to 100° C. o** to 300'' C. 1/582 V846 V68x 71x03 Vxo89 .17182 .18832 .11821 .14684 .08842 .09183 .206 .226 .142 .176 .106 .III .0115 .00418 .00788 .00326 .00589 .00204 LINEAR EXPANSION OF SOLIDS BY HEAT. 337 Table No. io8 {continued). GLASS. Flint Glass French Glass, with lead Glass tube, without lead Glass of St. Gobain Barometer tubes (Smeaton). Glass tube (Roy) Glass rod, solid (Roy) Glass (Dulong and Petit) . . . Do. (o° to 200° C.) Do. (o'' ta3oo°C.) Ice Expansion between 32* and 213* F. in common fractions. /1248 Am7 /1090 /zx2a /"75 /1289 /"37 V1032 V987 Expansion between 32* and 2X2° r. in a length =ioa .08117 .08720 .09175 .08909 .08333 .07755 .08083 .08613 .09484 .10108 Expansion between 32* and 2X2* F. in a length of xo feet. inch. .0974 .105 .110 .107 .100 .0931 .0970 .103 .114 .121 sion Expansi for X F. in a length of xco feet. inch. .00541 .00581 .00612 .00594 .00555 .00517 .00539 .00574 .00632 .00674 ■0333 STONES. Granite Do Cky-^late Do. York paving. Micaceous sandstone. Do. do. Do. do. Do. do. Do. do. Do. do. Carrara marble Sost do Stock Brick , Initial , Temperature. Final Temperature. 45^ 45 46 46 46 52 52 52 52 45 45 32 32 52 F. 220 100 87 104 95 200 200 150 100 100 260 212 212 260 F. Expansion in a length =xoo. length = too. .2916 .0416 .0416 .0693 .1695 .1736 .1041 .0832 .0520 .0416 .1458 .0849 .0568 .2500 Expansion for X F. in a length of xoo feet. inch. .0200 .00908 .0122 .0143 .0415 .0141 .00844 .0102 .01300 .00908 .00814 .00566 .00380 .00144 25" 338 HEAT. Speaking exactly, the cubical expansion is rather less than three times, and the superficial expansion tather less than twice, the linear expansion ; for, in fact, the expanded comers of the body are carried out to the full square figure, and have not the entering angles shown in the figure, and there is, in this way, a certain overlapping of the strata of expansion at the ends, sides, and top. The same kind of demonstration applies to bodies of any other than a cubical shape. A hollow body expands by heat to the same extent as if it were a solid body having the same exterior dimensions. The rate of expansion of solids from the freezing point to the boiling point of water, 32° to 212*' R, is sensibly uniform. The table. No. 108, gives the linear expansion of a number of metals, and of glass, between the freezing and boiling points;, and of ice for one degree, and of stones for various intervals of temperature. Authorities: — Laplace and Lavoisier, Smeaton, Roy, Troughton, Wollaston, Dulong and Petit, Froment, Rennie. Zinc is the most expansible of the metals; it expands fully one-third per cent, or as much as Vs^iSt part of its length, when heated from 32° F. to 212° F. Iron expands about one-seventh to one-eighth per cent.; and cast-iron and platinum about one-tenth per cent. The expansion of metals proceeds at a less rate above the boiling point than below it. Ice expands at the rate of V36,oooth of its length for one degree Fahrenheit; which, for 180°, would be Vaooth of its length, — greatly more than that of any metal. Expansion of Liquids. The measurement of the expansion of liquids by the application of heat cannot well be taken lineally; that is, as linear expansion, in th6 sense in which the expansion of solids is observed. For liquids must be con- tained in vessels, which only admit of expansion in one direction, seeing that the liquid is limited by the bottom and sides of the vessel, which throw the whole of the expansion or enlargement of volume upwards. The observations on the expansion of liquids, therefore, though measured in one direction only, necessarily indicate the cubical expansion or total enlargement of volume. But, of course, it is easy to reduce the expansion of a liquid for comparison with the linear expansion of a solid by taking one-third of the observed measurement. When the temperature of water at the freezing point, 32° F., is raised, the water does not at first expand, but, on the contrary, contracts in volume until the temperature is raised to 39°.! F., which is 7.1 degrees above the freezing point. This is called "the temperature of maximum density." From this point water expands as the temperature rises, until, at 46° F., it regains its initial volume, that is, the volume at 32° F. Thence, it con- tinues to expand until it reaches the boiling point, 212° F., under one atmosphere. Passing this point upwards, if the pressure be suitably increased, water continues to expand with a rise of temperature. The cubical expansion of water when heated from 32® to 212° F. is .0466; that is, the volume is increased from i at 32° F. to 1.0466 at 212** F. This expansion is rather more than 4^ per cent, or between '/aiSt and '/aad part of the volume at 32°. The expansion of water increases in a EXPANSION OF LIQUIDS BY HEAT. 339 Table No. 109. — Expan^on and Density of Pure Water, FROM 32** TO 390° F. (Calculated by means of Rankine's approximate formiila. ) Tempera- ture. Comparative Volume. Comparative Density. Density, or weight of X cubic foot. Weight of X gallon. Remarkable Temperatures. Fahr. Water at 32' = 1. Water at 32' = I. Pounds. Pounds. / 32^ 1. 00000 1. 00000 62.418 lO.OIOI Freezing point. 35 0.99993 1.00007 62.422 10.0103 39-1 0.99989 1. 000 1 1 62.425 IO.OII2 Point of maximum density. 40 0.99989 1. 0001 1 62.425 IO.OII2 45 0.99993 1.00007 62.422 10.0103 - 46 1. 00000 1. 00000 62.418 lO.OIOI f Same volume and density 1 as at the freezing point 50 1. 00015 0.99985 62.409 10.0087 (Weight taken for ordi- \ nary calculations. 52.3 1.00029 0.99971 62.400 10.0072 55 1.00038 0.99961 62.394 10.0063 60 1.00074 0.99926 62.372 10.0053 1 62 I.OOIOI 0.99899 62.355 10.0000 Mean temperature. 65 I.00II9 0.99881 62.344 9.9982 70 I.OOI60 0.99832 62.313 9-9933 75 1.00239 0.99771 62.275 9.9871 80 1.00299 0.99702 62.232 9.980 85 1.00379 0.99622 62.182 i 9-972 90 1.00459 0-99543 62.133 9.964 95 1.00554 0.99449 62.074 9-955 100 1.00639 0-99365 62.022 9-947 1 Temperature of conden- ( ser water. 105 1.00739 0.99260 61.960 9-937 no 1.00889 0.991 19 61.868 9.922 • "5 1.00989 0.99021 61.807 9-913 • 120 I.01I39 0.98874 61.715 9.897 125 1. 01239 0.98808 61.654 9.887 130 1. 01390 0.98630 61.563 9-873 135 I-OI539 0.98484 61.472 9-859 140 1.01690 0-98339 61.381 9.844 145 1. 01839 0.98194 61.291 9.829 150 1. 01989 0.98050 61.201 9.815 155 1. 02164 0.97882 61.096 9-799 160 1.02340 0.97714 60.991 9.781 165 1.02589 0.97477 60.843 9-757 170 1.02690 0.97380 60.783 9-748 17s 1.02906 0.97193 60.665 9-728 180 1.03 100 0,97006 60.548 9.711 185 1.03300 0.96828 60.430 9.691 340 HEAT. Table l^o, io<) {continued). (Calculated by means of Rankine's approximate formula.) Tempera- ture. Comparative Volume. Comparative Density. Density, or weight of I cubic foot. Weight of z gallon. Remarkable Temperatures. Fahr. Water at 32* = I. Water at 33* = I. Pounds. Pounds. 190 1.03500 0.96632 60.314 9.672 195 1.03700 0.96440 60.198 9.654 200 1.03889 0.96256 60.081 9.635 205 1. 0414 0.9602 59.93 9.61I 210 1.0434 0.9584 59.82 9.594 212 1.0444 0.9575 59.76 9-584 Boiling point; by formula j 212 1.0466 0.9555 59.64 9.565 r Boiling point; by direct ( measurement 230 1.0529 0.9499 59-36 9.520 250 1.0628 0.941 1 58.75 9.422 270 1.0727 0.9323 58.18 9.331 290 1.0838 0.9227 57.59 9.236 ( Temperature of steam of I 50 lbs. effective pres- ( sure per square inch. 298 1.0899 0.9175 57.27 9.185 I Temperature of steam of 338 I.III8 0.8994 56.14 • 9.004 < 100 lbs. effective pres- ( sure per square inch. i Temperature of steam of 366 I.I3OI 0.8850 55.29 8.867 < 150 lbs. effective pres- ( sure per square inch. i Temperature of steam of -| 205 lbs. effective pres- ( sure per square inch. 390 1. 1444 0.8738 54.54 8.747 greater ratio than the temperature. The annexed table No. 109 shows approximately the cubical expansion, comparative density, and comparative volume of water for temperatures between 32° and 212° F., calculated by means of an approximate formula constructed by Professor Rankine as follows : — D, nearly = ■? — ° — ^ / + 461 5 00 (I) 500 / + 461 in which 00=62.425 lbs. per cubic foot, the maximum density of water, and D, = its density at a given temperature / F. Rule. — To find approximately the density of water at a given temperaturCy the maximum density being 62.425 lbs, per cubic foot. To the given tempera- ture in Fahrenheit degrees, add 461, and divide the sum by 500. Again, divide 500 by that sum. Add together the two quotients, and divide 124.85 by the sum. The final quotient is the density nearly. EXPANSION OF LIQUIDS BY HEAT. 34 1 The results given by this rule are very nearly exact for the lower tempera- tures, but for the higher temperatures they are too great. For 212° F. the density of water by the rule is 59.76 lbs. per cubic foot, but it is actually only 59.64 lbs., showing an error of about Vsooth part in excess. From the table it appears that the density of water at 46° F., or about 8° C, is the same as at the freezing point, 32*^ F., and that the temperature of maximum density, 39°. i F., or 4° C., lies midway between those tempera- tures. The expansion of water towards and down to the freezing point is Vgoooth part of the volume at the temperature of maximum density. It would appear that in thus expanding from 39^.1 F. downwards, the particles of water enter on a preparatory stage of separation, anticipating the still further separation which ensues on the conversion of water into the solid state; for ice is considerably lighter than water and floats on it, and its density is little more than nine-tenths that of water. In passing upwards from the freezing point towards higher temperatures, the increase of volume of watef by expansion, in parts of the volume at the freezing point, is as follows : — Expansion in parts of the volume at 32' F. at 52°.3 F. corresponding to the weight per cubic foot (62.4 lbs.) usually taken for ordinary calcu- per cent latio ns .03 at 62° the mean temperature .10 at 1 00° the temperature of condenser water. .64 at 212° the boiling point 4.66 at 298° the temperature of steam of 50 lbs. effective pressure per square inch 9.0 at 338** the temperature of steam of 100 lbs. effective pressure per square inch 1 1.2 at 366° the temperature of steam of 150 lbs. effective pressure per square inch 13.0 at 390° the temperature of steam of 205 lbs. effective pressure per square inch 14.4 The expanded volume of some liquids from 32*^ to 212° F. is given in table No. no; that is, the apparent expansion as seen through glass. It is shown that alcohol and nitric acid are the most expansible, and water and mercury the least; the former expand one-ninth of their initial volume, and of the latter, water, as already stated, expands ^/aad part, and mercury Yesth part of their initial volumes respectively. Observations on the absolute expansion of mercury are added, and they show that whereas the apparent expansion in glass is 'As^h part, the real expansion is ^/^s^h part of the initial volume. No other liquid besides water has a point of maximum density ; that is, a point higher than the freezing point of the liquid. 342 HEAT. Table No. no. — Expansion of Liquids by Heat, from 3 2*^ to 212° F. Apparent Expansion, in Glass. Liquid. Alcohol Nitric Acid Olive Oil Linseed Oil Turpentine Sulphuric Ether Hydrochloric Acid (density 1.137) Sulphuric Acid (density 1.850) Water saturated with Sea Salt Water Mercury Volume at aia" F. volume at 3a* F.=i. I.IIOO I.IIOO 1.0800 1.0800 1.0700 1.0700 1.0600 1.0600 1.0500 1.0466 I.OI54 Ex]>ansion in Vulga^ Fractions. ■/, 18 volume at 33* F.=:i. } V. Vm Vm V.7 ■■',: •A 'As ao Absolute Expansion of Mercury. Volume at aia* F. Mercury, from 32* to 212** F. ( g** to 100" C), Dulpngand Petit, i. 01 80180 Do. from 212° to 392° F. (100* to 200' C), do. 1.0184331 Do. from 392" to 572° F. (200" to 300'* C), do. i. 0188679 Do. from 32** to 212" F. ( o" to 100' C), Regnault, 1.0181530 Expan- sion. V 55-5 V53 Expansion of Gases by Heat. Gases are divisible into two classes — permanent gases and vapours. Gases for which great pressure and extremely low temperatures are neces- sary to reduce them to the liquid form, are called permanent gases, and those which exist in the fluid state under ordinary temperatures, are called vapours. The influence of heat in expanding a permanent gas maintained under a constant pressure, is such that, for equal increments of temperature, the increments of volume by expansion are also equal or very nearly equal ; in Other words, the gas expands uniformly, or very nearly uniformly, in pro- portion to the rise of temperature. Again, it has been observed that when the volume of permanent gases is maintained constant, the pressure increases uniformly, or nearly uniformly, with an increase of temperature. A perfect or ideal gas is one which, under a constant pressure, expands with perfect uniformity in proportion to the rise of temperature; and of which, also, when confined to a constant volume, the pressure increases with per- fect uniformity in proportion to the rise of temperature. When the temperature of atmospheric air is raised from 3 2** to 212° F., the following are the total increments of volume or of pressure, according to the treatment, as determined by Regnault, when the volume at 32** is taken as i : — EXPANSION OF GASES BY HEAT. 343 Air. Tbmpsraturb. Incrbmsnt. Pressure constant 32° to 212" F Volume increased from i to 1.3670. Volume constant 32° to 212° F Pressure increased from i to 1.3665. Showing that the increase of pressure, .3665, with a constant volume, is sensibly the same as, though less than, the expansion or increase of volume, .3670, when the pressure is constant The table No. 1 1 1 gives the expansion and the increase of pressure, for several gases, when raised from 32° to 212'' F.: — Table No. in. — Expansion and Pressure of Gases raised from 32°t0 2I2'*F. (Renault.) Gasbs. Expannon of Gases under x Atmosphere. Increase of Pressure of Gases under a Con- tant Volume. Final Volimic at aia' F. Estpansion at sza* F., in Common Fractions. Fizud Pressure at 2x3*. Atmospheric Air Hvdroflfen Initial volume at 32*=!. 1.3670 1. 3661 1.3669 I.3710 I.3719 1-3877 13903 Initial volume at 39*= z. /a.73 ;A.73 /2.71 /a. 7a Va.6i Va.6o Initial pressure at 32*= X. 1.3665 1.3667 1.3668 1.3667 1.3688 1.3676 1.3829 1.3843 Nitrocren Carbonic Oxide Carbonic Acid Nitrous Oxide Cvanocen Sulphurous Acid Table No. 112. — Expansion of Gases raised from 32° to 212° F., under Different Pressures, these Pressures remaining Con- stant FOR EACH Observation. (Regnault.) Gas. Pressure. Volume at 2x2*. Air • Millimetres. 760 2525 2620 Atmospheres. I. GO 3.32 3.45 Volume at 32' F. = x. 1.36706 1.36944 1.36964 Hydrogen 760 2545 1. 00 3.35 I.36613 I.36616 Carbonic Acid 760 2520 1. 00 332 1.37099 1.38455 Sulphurous Acid 760 980 1. 00 1. 16 1.3903 1.3980 344 HEAT. The first part of the table, No. iii, on the expansion of gases by heat, shows that the expansion, which is a little more than a third of the initial volume, is nearly die same for air, hydrogen, and carbonic oxide, which are sensibly perfect gases, and have never been liquefied. On the contrary, carbonic acid, cyanogen, and sulphurous acid have a greater enlargement of volume than those gases, and they are gases which may easily be liquefied. The second part of the table, column 4, shows that, when the volume is constant, the pressure is increased nearly in the same proportion as the volume is increased, when the pressure is constant. This nearness of the proportions is particularly close in the cases of the three sensibly perfect gases, — ^air, hydrogen, and carbonic oxide. The next table. No. 112, contains the results of Regnault's experiments on the expansion of gases from 32° to 212° F., under various constant pressures of from i to 3^ atmospheres. It is shown that the expansions of air and of hydrogen are sensibly the same, whether the constant pressure be I atmosphere or between 3 and 4 atmospheres ; whilst the expansions of carbonic acid and sulphurous acid are higher at the higher pressure. The deductions of Regnault, from his experiments, comprised the following principles : — That for air, and all other gases except hydrogen, the coeffifcient of dilatation, or the increment of expansion for one degree rise of temperature, increases to some extent with their density. That all gases possess the same coefficient of dilatation when in a state of extreme tenuity ; but that this law is departed from as gases become dense. Adopting, nevertheless, the mean of the results of the experiments of M. Regnault and of M. Rudberg, the expansion of one volume of air measured at 32** F., when heated to 212° F., under a constant pressure, will, for future calculation, be taken as equal to 0.365; the ratio of the initial to the expanded volume being as i to 1.365. As the expansion is uniform with the rise of the temperature through 180°, the expansion for each degree Fahr. is — .365-^180 = —^ - , 493-2 the volume at 32° F. being = i. The same uniform rate of expansion holds sensibly for temperatures higher than 212°; it has been verified experi- mentally up to 700° F., under one atmosphere. It is inferred that, con- versely, air would contract uniformly under uniform reductions of temperature below 32^* F., until, on arriving at 493°. 2 below the freezing point, or 46 1**. 2 F. below zero, the air would be reduced to a state of collapse, without elasticity. This point in the Fahrenheit scale has thus been adopted as that of absolute zero, standing at the foot of the natural scale of temperature; and the temperature, measured from absolute zero, or — 461^2 F., is called the absolute temperature. Accordingly, if a given weight of air at 0° F. be raised in temperature to + 461° F., under a constant pressure, it is expanded to twice its original volume; and if heated from 0° F. to twice 461°, or 922°, its original volume is trebled. In briefi ^^ follows that, sensibly. EXPANSION OF GASES BY HEAT. 345 ist The pressure of air varies inversely as the volume when the tempera- ture is constant 2d. The pressure varies directly as the absolute temperature when the volume is constant. 3d. The volume varies as the absolute temperature when the pressure is constant. 4th. The product of the pressure and volume is proportional to the absolute temperature. The absolute zero-point by different thermometrical scales is as follows : — Reaumur - 2 1 9°. 2 Centigrade -274° Fahrenheit - 46i°.2 To simplify calculation, the decimal is usually dropped from the Fahrenheit temperature, which is taken as -461°. The foregoing laws do not apply exactly to the expansion and contraction of the more easily condensable gases, for these, as they approach the point of liquefaction, become sensibly more compressible than air. Oxygen, nitrogen, hydrogen, nitric oxide, and carbonic oxide follow the same ratio of compression as that of air, being incondensable gases, at least as far as 100 atmospheres of pressure. Sulphurous acid, ammoniacal gas, carbonic acid, and protoxide of nitrogen, which have been proved, on the contrary, to be condensable, become sensibly more compressible than air when they are reduced to one-third or one-fourth of their original volume at atmospheric pressure. Carbonic acid, under five atmospheres, occupies only 97 per cent of the volume which air occupies under the same pressure ; and under forty atmospheres, near the condensing point, it occupies only 74 per cent, or barely three-fourths of the volume of air at the same pressure. It has, nevertheless, been established that all gases, at some distance from the point of maximum density for the pressure, beyond which point they must condense, sensibly follow the first law above recited, according to which the pressure and the density vary directly as each other, when the temperature is constant With such limitations, they rank as perfect gases. The table No. 113 contains examples of the progressive pressures required to compress air, nitrogen, carbonic acid, and hydrogen, into one-twentieth of their original volumes, founded on experiments made by M. Regnault The pressures are expressed in metres of mercury, the pressure of a column of mercury one metre high being equal to 19.34 lbs. per square inch. The table shows that hydrogen is the most perfect type of gaseity. When compressed to a twentieth of its original volume, it supports something more than twenty times the original pressure. Air, on the contrary, requires a quarter of a metre less than 20 metres of pressure ; nitrogen requires a fifth of a metre less; and carbonic acid, like an overloaded spring, 3^ metres less. 346 HEAT. Table No. 113. — Compression of Gases by Pressure under a Constant Temperature. 1 Ratio of the Pressiuv in Metres of Mercury for original volume to the reduced volume. Air. Nitrogen. Carbonic Acid. Hydrogen. Metres. Metres. Metres. Metres. I 1.000 1. 000 1. 000 1. 000 2 1.998 1.997 1.983 2.001 4 3-987 3.992 3.897 4.007 6 5-970 5.980 5-743 6.018 8 7.946 7.964 7-519 8.034 10 9.916 9-944 9.226 10.056 12 11.882 II.919 10.863 12.084 14 13-845 13.891 12.430 I4.II9 16 15.804 15.860 13.926 16.162 18 17-763 17.825 15.351 18.2II 20 19.720 19.789 16.705 20.269 Note, — 20 metres of mercury are equal to a pressure of 386.8 lbs. per square inch, or 26.3 atmospheres. Relations of the Pressure, Volume, and Temperature of Air and other Gases. In accordance with the relations of pressure, volume, and temperature above stated, it is found that air and other perfect gases, and, within practical limits, the permanent gases generally, are expanded by heat at the rate of V461 part of their volume at 0° F. for each degree of temperature, under a constant pressure. If the volume at the freezing point, 32° F., be taken as the point of departure, the denominator of the fraction is 461° + 32° = 493°, and the expansion is at the rate of ^453 part of the volume at 32° F. for each degree of temperature. In general, for any- other initial temperature the denominator of the fraction showing the rate of expansion for each degree is found by adding 461° to the initial tempera- ture. But, for convenience of calculation, the initial temperature is usually- taken at 0° F. Similarly, the pressure of air having a given constant volume, is increased by heat at the rate of 7461 part of the pressure at 0° F. The fraction of expansion when the pressure is constant, and the fraction of pressure when the volume is constant, for each degree of temperature by Fahrenheit's scale above o^, is, then. 461' and the same fraction expresses the rate of contraction of volume for each degree of temperature below o® F. A number of proportions and rules for the relations of the pressure, volume, and temperature of a constant weight of a gas are readily deduced from the above defined ratios. PRESSURE, ETC., OF AIR AND OTHER GASES. 347 1. When the pressure is constant, the volume varies as the absolute temperature; or, V : V: : / + 461 : /' + 461, and v/_v ^ + 461 . / \ " 7T^' ^^ in which V is the volume of the air or other gas at the temperature /, and V is the volume at the temperature /^. Whence the rule — Rule i. To fi?id the volume of a constant weight of air or otJier permanent gas, at any other temperature, when the volum^ at a given temperature is known, the pressure being constant. Multiply the given volume by the new absolute temperature, and divide by the given absolute temperature. The quotient is the new volume. Note, — The absolute temperature is found by adding 461° to the temperature indicated by the Fahrenheit thermometer. A common case of the above rule is, air at the mean temperature, 62** F., and mean atmospheric pressure, 14.7 lbs. per square inch. The increased volume, by expansion by heat, taking the initial volume = i, is found by substitution and reduction to be as follows : — v'jy^^2 (2) 523 Rule 2. To find the increased volume of a constant weight of air, of which the initial volume =1, taken at 62° F., heated under the ordinary atmospheric pressure of 14.7 lbs. per square inch, to a given temperature. To the given temperature add 461, and divide the sum by 523. The quotient is the increased volume by expansion. 2. When the temperature of a constant weight of air, or other gas, is constant, the volume varies inversely as the pressure ; or, V : V : : /' : /, and V' = V^; (3) in which V and V are the volumes respectively at the pressures/ and/'. Rule 3. To find the volume of a constant weight of air or other permanent gas, for any pressure, when the volume at a given pressure is known, the temperature remaining constant. Multiply the given volume by the given pressure, and divide by the new pressure. The quotient is the new volume. 3. When the pressure and temperature of a constant weight of air or other gas both change, the volume varies in the compound ratio of the absolute temperature directly, and the pressure inversely ; or, V : V ::/(/ + 46i):/(/' + 46i); or V'/ (/ + 46i) = V/ (/'+461), and V/-.v /(^ + 46i) , V 34^ HEAT. Rule 4. To find the volume of a coftstant weight of air or other permanait gas for any other pressure and temperature, when the volume is known at a given pressure and temperature. Multiply the given volume by the given pressure, and by the new absolute temperature, and divide by the new pressure, and by the given absolute temperature. The quotient is the new volume. 4. When the volume and temperature of a constant weight of air or other gas both change, the pressure varies in the compound ratio of the absolute temperature directly, and the volume inversely. / :/ : : V'(/ + 46i) : V(/' + 46i); or V'/(/ + 46i) = V/ (/' + 461), and ^ ^V'(/+46i) ^^^ Rule 5. To find the pressure of a constant weight of air or other permanent gas for any other volume and temperature, when the pressure is known for a given volume and temperature. Multiply the given pressure by the given volume, and by the new absolute temperature, and divide by the new volume, and by the given absolute temperature. The quotient is the new pressure. For the common case, when the initial temperature is 62® F., and the initial pressure is 14.7 lbs. per square inch, the formula (5) becomes, by substitution and reduction, v=VJ^+_46xi (6) ^ 35-58 V' ^ > Rule 6. To find the pressure of a constant weight of air or other gas taken at 62° F., and at 14.7 ibs, pressure per square inch, with a given volume, for any other volume and temperature. Multiply the initial volume by the final temperature plus 461, and divide the product by the final volume, and by 35.58. The quotient is the new pressure in lbs. per square inch. When the volume is constant, with an initial temperature of 62** F., and an initial pressure of 14.7 lbs. per square inch, the above formula (6) is simplified thus : — p'=(±^ ^ J ^ 35-58 ^^' Rule 7. To find the pressure of a constant weight of air or other gas takefi at 62° F., and at 14.7 lbs. pressure per square inch, with a constant volume, for a given temperjiture. Add 461 to the given temperature, and divide the sum by 35.58. The quotient is the pressure in lbs. per square inch. 5. The mutual relations of pressure, volume, and temperature are con- densed in the following formula : — V/*/ + 46i, {a) the product of the volume and pressure of a constant weight of air being proportional to the absolute temperature. And, as that product bears always the same ratio to the absolute temperature, an equation may be PRESSURE, ETC., OF AIR AND OTHER GASES. 349 formed between them by multiplying the absolute temperature by a coefficient, which may be put = a. Then — V/ = ^ (/+461); {b) that is, the product of the volume and pressure of a constant weight of air or other permanent gas, is equal to the absolute temperature multiplied by a constant coefficient, which is to be determined for each gas according to its density. Special Rules for One Found Weight of a Gas. The application of formula {b) to a particular constant weight of gas, will suffice for many purposes. Let the constant weight be one pound of gas. To settle the coefficients for the different gases, take, for example, the temperature 32° F., giving an absolute temperature of 493**, and the pressure one atmosphere, or 14.7 lbs. per square inch. The volume of one pound of air at this temperature and this pressure is as before stated, 12.387 cubic feet Substitute these values for V, /, /, in the formula (^), then — 12.387 X 14.7 =ax 493, whence the coefficient, a, for air is — ^ = -36935* or 2.7074 and the formula {b) becomes, for air, / + 461. V/ = 2.7074 (O Table No. 114. — Of Coefficients or Constants, «, in the Equation {b) FOR THE Relations of the Volume, Pressure, and Tem- perature OF Gases; namely, ^ p - a (/ + 461). > Name of gas. Volume of one pound of gas, at 32** F., under one atmosphere. Value of coefficient a. Hydrogen Gaseous steam cubic feet. 178.83 19-913 12.753 12.580 12.387 11.205 8.157 8.IOI 4.777 1.776 5-33200, or 7o.x875 0.59372, or 7^.6842 0.38027, or 72.6297 0.37506* or 7a.6662 0.36935* or 72.7074 0.33406, or 7a.5535 0.24322, or 74.„x4 0.24155, or 74.,3p9 0.14246, or 77.019s 0.05296, or 7,8.878 Nitrogen defiant gas Air Oxygen Carbonic acid (ideal)* * . Do. do. (actual) Ether vapour* Vapour of mercury* * The densities are computed by Rankine for the ideal condition of perfect gas. 3SO HEAT. that is to say, the volume of one pound of air, multiplied by the pressure per square inch, is equal to the absolute temperature divided by the constant 2.7074. To adapt the formula (d) for other gases, the respective coeflScients, or constants, are found in the same manner, in terms of the volume of one pound of each gas, at 32° F., under one atmosphere of 14.7 lbs. per square inch. They are given in table No. 114. 6. The volume of one pound of air at any pressure and temperature is deduced as follows : — V=^+46i ^gj 2.7074/ ^ ' Rule 8. — To find the volume of ofie pound of air, of a given temperature and pressure. Divide the absolute temperature by the pressure in lbs. per square inch, and by 2.7074. The quotient is the volume in cubic feet. For the ordinary case When the pressure is constant at 14.7 lbs, per square inch, the formula (8) becomes, by substituting and reducing, ^ = lPo- ^9) Rule 9. — To find the volume of one pound of air under 14.7 lbs, pressure per square inch, at a given temperature. Add 46 1 to the temperature, and divide the sum by 39.80. The quotient is the volume in cubic feet. 7. The pressiure of one pound of air of any volume, and at any tempera- ture, is found as follows : — ^ = ^7^4V <'°) Rule 10. — To find tJie pressure of one pound of air, of a given temperature and volume. Divide the absolute temperature by the volume and by 2.7074. The quotient is the pressure in lbs. per square inch. 8. The temperature of one pound of air of any volume and pressure is found as follows : — /= 2.7074 V/- 461 (11) Rule i i. — To find the temperature of one pound of air, of a given relume and pressure. Multiply the volume by the pressure in pounds per square inch, and also by 2.7074; subtract 461 from the product. The remainder is the temperature. 9. The density of air is inversely as the volume, and is expressed by an inversion of the formula (8), for the volume; thus, putting D for the density, or the weight in pounds of one cubic foot of air — ^ = '-7°747:j^ (") Rule 12. — To find the density of air, at a given temperature and pressure. Multiply the pressure in pounds per square inch by 2.7074, and divide by the absolute temperature. The quotient is the density, or weight in pounds of one cubic foot. VOLUME, DENSITY, AND PRESSURE OF AIR. 351 Table No. 115. — Volume, Density^ and Pressure of Air at various Temperatures. 1 1 Volame of one pound of air at Density, or weight Pressure of a given weight of air having a constant volume. Temperature. constant atmospheric pressure. of one cubic foot of 14.7 lbs. per square inch. Daium— Volume at 62"' F. = i. air at atmospheric Datum — Atmospheric pressure at 62- f: = X. pressure. Fahrenheit. cubic feet. comparative volume. pounds. pounds per square inch. comparative pressure. 0^ .11-583 .881 .086331 12.96 .881 32 12.387 •943 .080728 13.86 -943 40 12.586 -958 .079439 14.08 -958 50 12.840 -977 .077884 14.36 -977 62 13-141 1. 000 .076097 14.70 1. 000 70 13-342 1.015 .074950 14.92 1.015 80 13.593 1.034 .073565 15.21 1.034 90 13-845 1-054 .072230 15-49 1.054 ICO 14.096 1.073 .070942 15.77 1-073 120 14-592 I. Ill .068500 16.33 I. Ill 140 15.100 1.149 .066221 16.89 1.149 160 15-603 1. 187 .064088 17.50 1. 187 180 16.106 1.226 .062090 18.02 1.226 200 16.606 1.264 .060210 18.58 1.264 210 16.860 1.283 -059313 18.86 1.283 212 16.910 1.287 .059135 18.92 1.287 220 I7.III 1.302 .058442 19.14 1.302 230 17.362 1.321 -057596 19.42 1. 321 240 17.612 I -340 .056774 19.70 1.340 250 17.865 1-359 .055975 19.98 1-359 260 18.I16 1-379 .055200 20.27 1-379 270 18.367 1.398 -054444 20.55 1.398 280 18.621 1.417 -053710 20.83 1.417 290 18.870 1.436 .052994 21. II 1.436 300 I9.I2I 1-455 .052297 21.39 1-455 320 19.624 1-493 .050959 21.95 1-493 340 20.126 1-532 .049686 22.51 1-532 360 20.630 1-570 .048476 23.08 1.570 380 21. 131 1.608 .047323 23-64 1.608 400 21.634 1.646 .046223 24.20 1.646 425 22.262 1.694 .044920 24.90 1.694 450 22.890 1.742 .043686 25.61 1.742 475 23-518 1.789 .042520 26.31 1.789 500 24.146 1.837 .041414 27.01 1-837 525 24.775 i.«85 .040364 27.71 •1.885 550 25-403 1-933 .039365 28.42 1.933 575 26.031 1.981 .038415 29.12 1.981 600 26.659 2.029 .037510 29.82 2.029 352 HEAT. • Table No. 115 {continued). Volume of one pound of air at Density, or weight Pressure of a given weight of air having a constant volume. constant atmospheric pressure. of one cubic foot of Temperature. 14-7 lbs. per square inch. Datum — ^Volume at 62" F. = i. air at atmospheric Datum — Atmosc 6a' F. thenc pressure at pressure. = X. Fahrenheit. cubic feet. comparative volume. pounds. pounds per square inch. comparative pressure. 650 27.915 2.124 .035822 31.23 2.124 700 29.172 2.220 .034280 32.63 2.220 750 30.428 2.315 .032865 34.04 2.315 800 31.685 2.41 1 .031561 35.44 2.41 1 850 32.941 2.507 .030358 36.85 2.507 900 34.197 2.602 .029242 38.25 2.602 950 35.453 2.698 .028206 39.66 2.698 1000 36.710 2.793 .027241 41.06 2.793 1500 49.274 3.749 .020295 55.12 3.749 2000 61.836 4.705 .016172 69.17 4.705 2500 74.400 5.661 .013441 83.22 5.661 3000 86.962 6.618 .011499 97.28 6.618 Note to Rules 8, 9, 10, 11, 12. — ^The coefficients or constants for other gases, in the application of the preceding five formulas and rules, are given in table No. 114. The table No. 115 contains the volume, density, and pressure of air at various temperatures from 0° to 3000° F., starting from 62° F. and 14.7 lbs. per square inch respectively as unity for the proportional volumes and pres- sures. The second column of the table, containing the volumes of one pound of air at different temperatures, was calculated by means of the formula (9), page 350. The third column, of comparative volumes, the volume at 62° F. being = i, was calculated by means of formula (2), page 347. The fourth column, of density, contains the reciprocals of the volumes in column 2, but it is calculable independently by means of formula (12), page 350. The fifth column, of pressures, due to the temperatures, was calculated by means of formula (7), page 348. The sixth column contains the pressures expressed comparatively, the atmospheric pressure, 14.7 lbs. per square inch, being taken as i. SPECIFIC HEAT. The specific heat of a body signifies its capacity for heat, or the quantity of heat required to raise the temperature of the body one degree Fahrenheit, compared with that required to raise the temperature of a quantity of water of equal weight one degree. The British unit of heat is that which is required to raise the temperature of one pound of water one degree, from 32° F. to 33** R; and the specific heat of any other body is expressed by the quantity of heat, in units, necessary to raise the temperature of one pound weight of such body one degree. The specific heat of water is represented by i, or unity, and there are very few bodies of which the specific heat equals or exceeds that of water. Specific heats are, therefore, almost universally expressible by firactions of a unit. SPECIFIC HEAT OF WATER. 353 It is necessary to fix a standard of temperature, such as the freezing point, for the datum of specific heat, as the specific heat of water is not exactly the same at different parts of the scale of temperatures, but increases in an appreciable degree, as well as in an increasing ratio, as the tempera- ture rises. For temperatures not higher than 80° or 90° F., the quantity of heat required to raise the temperature of water one degree is sensibly constant ; at 86° F., it is not above one-fifth per cent, in excess of that at the freezing-point. At 212° F., it is about \y^ per cent, in excess of that at 32° F. Above 212° F., it increases more rapidly; at 302% it is 2^ per cent, more than at 32°, and at 402°, it is 4^^ per cent. more. The average specific heat of water between the freezing and the boiling points is 1.005, or one-half per cent, more than the specific heat at the freezing point. It follows from the increasing specific heat of water, as the temperature rises, that the consumption of heat in raising the temperature is slightly greater expressed in units than in degrees of temperature. To raise, for example, one pound of water from 0° to 100° C, or from 32** to 212** F., there are required 100.5 ^- units, or 180.9 ^' units, of heat. The specific heats of water in the solid, liquid, and gaseous state are grouped as follows : — Ice 0.504 Water i . 000 Gaseous Steam 0.622 showing that in the solid state, as ice, the specific heat of water is only half that of liquid water ; and that, in the gaseous state, it is a little more than that of ice, or barely five-eighths of that of liquid water. The specific heat of all liquid and solid substances is variable, increasing sensibly as the temperature rises, and the specific heats of such bodies, as tabulated, are not to be taken as exact for all temperatures, but rather as approximate average values, sufficiently near for practical purposes. The specific heat of the same body is, however, nearly constant for temperatures under 212° F. The specific heats of such gases, on the contrary, as are perfectly gaseous, or nearly so, do not sensibly vary with density or with temperature. For the same body, the specific heat is greater in the liquid than in the solid state. For example : — Liquid. Solid. Water (specific heat) i .000 o. 504 Bromine „ o.iir 0.084 Mercury „ 0.0333 0.0319 M. Regnault has verified, by numerous experiments, the conclusion arrived at by previous experimentalists, that, for metals, the specific heats are in the inverse ratio of their chemical equivalents. Consequently the products of the specific heats of metals, by their respective chemical equivalents, are a constant quantity. The same rule holds good for other groups of bodies of the same composition, and of similar chemical constitu- tion. The specific heat of alloys is sensibly equal to the mean of those of the alloyed metals. The following are the specific heats of water for various tempera- 88 -thermometer, calculated . (i) tures from o° to 230° C, or 32° to 446° F., by the a by means of Regiiault's fonnula : — c- I + 0.00004 /+ 0.0000009/-; in which ^ is the specific heat of water at any temperature /, the specific heat at the freezing point being - .1. Table No. 116, — Specific Heat of Water. 01 Specific 1 Units of H«t fie Hon Bl He romihefnieiLng , I e given thefr. poinl 10 I he giv prrature. anc KIHUIK. F^n=„b.i.. Cecil, unki. Fahr.uml^ F™ ngpoLnI=.. 3!° 0,000 0.000 I 0000 ^t 10.002 18.004 ' 0005 1 0002 68 aO.oio 36.018 OD12 I 0005 86 30.026 54.047 1 0020 I 0009 104 40,051 72.090 1 0030 I 0013 122 50.087 90.157 I 0042 1 0017 140 60.137 108.247 I 0056 0023 IS« 70.210 126.378 1 0072 1 0030 176 80.282 144.508 1 0089 1 0035 194 90.381 162.686 I 0109 1 0042 ai2 100.500 180.900 I 0130 1 0050 230 110.64. 199.152 I 0153 I 0058 248 120.806 217.449 I 0177 1 0067 266 130.997 235-791 1 0204 I 0076 284 141.21S 254-187 I 0232 1 0087 302 151.462 272.628 1 0262 I 0097 320 161.741 291.132 1 0294 I 0109 338 172.052 309.690 I 0328 I 356 182.398 328.320 I 0364 1 0133 374 192.779 347.004 1 0401 1 0146 39* 203,200 365.760 I 0440 1 0160 410 213.660 384-588 0481 I 0174 428 224.162 403.488 1 0524 1 0.89 446 234.708 422,478 I 0568 1 0204 The Specific Heat of Air and other Gases, :ific heat, or capacity for heat, of permanent gases is sensibly r all temperatures, and for all densities. That is to say, the heat of each g;as is the same for each degree of temperature. . Regnault proved that the capacity for heat was uniform for :s varying from -3o°C.to + 225°C. (- 22° to 437° F.); thus : heat for equal weights of air, at constant pressure, were as SPECIFIC HEAT OF AIR, ETC. 355 Air between -30"* and + 10° C Specific heat, 0.2377 Do. 10° and+ioo°C Do. 0.2379 Do 100° and + 225"* C Do. 0.2376 Average 0.2377 The temperature is then without any^ sensible influence on the specific heat of air; neither has the pressure, so far as it has been subjected to experiment — from one to ten atmospheres — any influence on the magni- tude of the specific heat The specific heat of gases is to be observed from two points of view: — I St, When the pressure remains the same, and the gas expands by heat. ' 2d, When the volume remains the same, and the pressure increases with the temperature. There is a striking diflference in the specific heat, or capacity for heat, according as it is measured under an increasing volume, or an increasing pressure. When the temperature is raised one degree, under constant pressure, with increasing volume, the gas not only becomes hotter to the same extent as when the volume remains the same and the pressure alone is increased, but it also expands 7493^ P^"^ ^^ ^^ volume at 32° F., and thus absorbs an additional quantity of heat in proportion to the work done by expansion against the pressure. It follpws that the specific heat of a gas at constant pressure is greater than that of the same gas under a constant volume; and though the former alone has been made the subject of direct experiment, the latter being of a diflicult nature for experimenters, yet the latter, which is properly the specific heat, is easily deducible from the former on the principle of the mechanical theory of heat When the volume of a gas is enlarged by expansion against pressure, the work thus done in expanding the gas may be expressed in foot-pounds by multiplying the enlargement of volume in cubic feet by the resistance to expansion in pounds per square foot Having thus found the work done in foot-pounds, it may be divided by Joule's equivalent, 772, and the quotient will be the expression of that work in units of heat. It becomes latent, or insensible to the thermometer, and is called the latent heat of expansion. It constitutes an expenditure of heat in addition to the heat that is sensible to the thermometer, and that raises the temperature. The sum of these two quantities of heat is that which has been observed in the gross by experimentalists, and which gives the specific heat at constant pressure. It follows that, when the specific heat at constant pressure is known, the specific heat at constant volume may be arrived at by subtracting the pro- portion of heat devoted to the enlargement of the volume from the total heat absorbed at constant pressure. The remainder is the proportion of heat necessary and sufficient to elevate the temperature when the volume remains unaltered, from which the specific heat at constant volume is deduced by simple proportion; thus — As the total heat absorbed at constant pressure, Is to the proportion of heat absorbed at constant volume, So is the specific heat at constant pressure To the specific heat at constant volume. For example, the specific heat of air at constant pressure and with in- I 356 HEAT. creasing volume has been observed to be .2377, that of water being i. Let one pound of air at atmospheric pressure, and at 32° F., having a volume equal to 12.387 cubic feet, be expanded by heat to twice its initial volume, the pressure remaining the same. The absolute temperature, which is 32° + 461 = 493** R, will be doubled, and the indicated temperature will be 32 + 493 = 525° F. Thus, 493 degp-ees of heat are appropriated, and if the capacity for heat of the air were the same as that of water, 493 units of heat would be expended in the process of doubling the volume. But, as the specific heat is only .2377, or less than a fourth of that of water, the expen- diture of heat is just 493 x. 23 77=117. 18 units, and this quantity comprises the fraction of heat consumed in displacing the atmosphere and overcoming its resistance through a space of 12.387 cubic feet additional to the original or initial volume of the same amount. Now, the work thus done is equal to— 12.387 cubic feet x 21 16.4 lbs. pressure per sq. foot = 26,216 foot-pounds; and dividing this by 772, Joule's equivalent, the work of enlarging or doub- ling the volume is found to be equivalent to 33.96 units of heat Deduct- ing these 33.96 units fi:om the gross expenditure, which is 11 7. 18 units, the remainder, 83.22 imits, is the proportion of heat required to raise the temperature through 493 degrees, under an increasing pressure simply, without increasing the volume ; and this remainder is the datum from which the proper specific heat of air is to be deduced. The distribution of heat thus detailed may be concisely exhibited thus : — Units. To double the temperature without adding to the volume.... 83.22 To double the volume, in addition 33'9^ To double the temperature and double the volume at con- stant pressure 117. 18 Now, as before stated, the specific heat at constant volume bears the same ratio to that at constant pressure, as the respective quantities, or units of heat, absorbed, do to each other, or as 83.22 and 11 7. 18; and it is found by simple proportion to be .1688 ; thus — 117. 18 : 83.22 : : .2377 : .1688. The proper specific heat of air is then .1688, in raising the temperature without enlarging the volume, and it bears to the so-called specific heat of air, at constant pressure and with expanding volume, the ratio of i to 1.408. This ratio, i to 1.408, deduced by means of the mechanical theory of heat, is practically identical with the ratio experimentally arrived at by M. Masson from the fall of temperature of a quantity of compressed air, which was liberated and allowed to expand back until it regained its initial pres- sure. The ratio he deduced firom his inverse experiment was i to 1.41; which is the ratio of I to \/3. SPECIFIC HEAT OF AIR, ETC. 357 It may be added, by way of explanation, and to enforce the distinction, that though the pressure of a gas under constant volume rises with the tem^rature, — a. phenomenon which is analogous, at first sight, to that of the volume of a gas at constant pressure increasing with the temperature, — yet there is no expenditure of work in simply raising the pressure in the former case, while the volume remains unaltered ; whereas, in the latter case, there is an expenditure in increasing the volume, as has already been shown. To generalize the foregoing process, by which the specific heat of air at constant volume has been deduced from the specific heat for constant pressure; and to show its applicability for finding the specific heat of all gases at constant volume : — Given / = the initial temperature of the gas, in degrees Fahrenheit. „ /' = the final temperature to which the gas is raised. V = the initial volume of the gas, under one atmosphere of pres- sure, in cubic feet. V = the final volume of the gas, heated under constant pressure. „ A = the specific heat of the gas under constant pressure. Put A' = the specific heat of the gas under constant volume. H = the total heat expended at constant pressure, in units of heat. H' = the total heat expended at constant volume. H^ = the fractional quantity of heat expended in increasing the volume, at constant pressure; or the latent heat of expansion. To find the value of A'; then by proportion. And >4' = -^j-. NowH' = H-H^ And — = — ^ — , and, by substitution, Xx xl >> » >^=<2^' (a) Again, H = (Z' - /) x ^, And H^ = {V-t/) X 14.7 X 144-^772 = (V-z/)x 2.742; And-^ Jjr^) • Substituting this value in equation (a) above, ^, >4 (>4 (/'-/)- 2. 742 (V-z>)) . * A (/' - /) ' ^^^,J{t'-t)-2^i_y{y-v) (^) Whence the following rule : — Rule i. To find the specific heat of a gas at constant volume, when the specific heat at constant pressure is given together with the initial and final temperatures due to given initial and final volumes under an atmosphere of 3S8 HEAT. pressure. Multiply the difference of the initial and final temperatures by the specific heat at constant pressure. Also, multiply the difference of the initial and final volumes by 2.742. Find the difference of these two pro- ducts, and divide it by the difference of the temp